State Whether Each Sentence Is True Or False . If False, Replace the Underlined Term to Make a True Sentence

Home , Incenter

State whether each sentence is true or false . If false, replace the underlined term to make a true sentence. 1. The altitudes of a triangle intersect at the centroid. SOLUTION: The centroid is the the point where the medians intersect. The orthocenter is the point where the altitudes intersect. false; orthocenter

ANSWER: false; orthocenter

2. The point of concurrency of the medians of a triangle is called the incenter. SOLUTION: The point where the medians intersect is the centroid. The point of concurrency of the angle bisectors of a triangleis called the incenter. The sentence is false. "The point of concurrency of the angle bisectors of a triangle is called the incenter." is the true sentence.

ANSWER: false; angle bisectors 3. The circumcenter of a triangle is equidistant from the vertices of the triangle. SOLUTION: The point that is equidistant from the vertices of a triangle is called the circumcenter. The statement is true.

Chapter 5 Study Guide and Review ANSWER: false; medians State whether each sentence is true or false . If 5. The perpendicular bisectors of a triangle are false, replace the underlined term to make a concurrent lines. true sentence. SOLUTION: 1. The altitudes of a triangle intersect at the centroid. The perpendicular bisectors of a triangle are SOLUTION: concurrent lines. The statement is true. The centroid is the the point where the medians intersect. The orthocenter is the point where the ANSWER: altitudes intersect. false; orthocenter true

ANSWER: 6. A proof by contradiction uses indirect reasoning. false; orthocenter SOLUTION: Indirect reasoning is key when writing a proof by 2. The point of concurrency of the medians of a triangle contradiction. The statement is true. is called the incenter. ANSWER: SOLUTION: true The point where the medians intersect is the centroid. The point of concurrency of the angle 7. A median of a triangle connects the midpoint of one bisectors of a triangleis called the incenter. The side of the triangle to the midpoint of another side of sentence is false. "The point of concurrency of the the triangle. angle bisectors of a triangle is called the incenter." is the true sentence. SOLUTION: A median of a triangle connects the vertex to the ANSWER: midpoint of the side opposite it. The sentence is false. false; angle bisectors The true sentence is "A median of a triangle connects the midpoint of one side of the triangle to 3. The circumcenter of a triangle is equidistant from the the vertex opposite that side." vertices of the triangle. ANSWER: SOLUTION: false; the vertex opposite that side The point that is equidistant from the vertices of a triangle is called the circumcenter. The statement is 8. The incenter is the point at which the angle bisectors true. of a triangle intersect. ANSWER: SOLUTION: true The point where the angle bisectors intersect is called the incenter. The statement is true. 4. To find the centroid of a triangle, first construct the angle bisectors. ANSWER: true SOLUTION: To find the centroid of a triangle, first construct the 9. Explain how to write a proof by contradiction. medians. The sentence is false. The true sentence is "To find the centroid of a triangle, first construct the SOLUTION: medians." Assume that the conclusion is false and show that this assumption leads to a statement that cannot be ANSWER: true. false; medians ANSWER: 5. The perpendicular bisectors of a triangle are Assume that the conclusion is false and show that concurrent lines. this assumption leads to a statement that cannot be true. SOLUTION: eSolutionsThe Manualperpendicular- Powered bisectorsby Cognero of a triangle are 10. Explain how to locate the largest angle in a scalenePage 1 concurrent lines. The statement is true. triangle. Then explain when a triangle does not have one largest angle. ANSWER: true SOLUTION: The largest angle in a scalene triangle is opposite the 6. A proof by contradiction uses indirect reasoning. longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third SOLUTION: angle, so the sides opposite the congruent angles are Indirect reasoning is key when writing a proof by longer than the base. In an equilateral triangle, all contradiction. The statement is true. angles are the same size.

ANSWER: ANSWER: true The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be 7. A median of a triangle connects the midpoint of one two congruent angles that are larger than the third side of the triangle to the midpoint of another side of angle, so the sides opposite the congruent angles are the triangle. longer than the base. In an equilateral triangle, all SOLUTION: angles are the same size. A median of a triangle connects the vertex to the midpoint of the side opposite it. The sentence is false. 11. Find EG if G is the incenter of . The true sentence is "A median of a triangle connects the midpoint of one side of the triangle to the vertex opposite that side." ANSWER: false; the vertex opposite that side SOLUTION: 8. The incenter is the point at which the angle bisectors By the Incenter Theorem, since G is equidistant from of a triangle intersect. the sides of , EG = FG. Find FG using the Pythagorean Theorem. SOLUTION: The point where the angle bisectors intersect is called the incenter. The statement is true.

ANSWER: true

9. Explain how to write a proof by contradiction. SOLUTION: Since length cannot be negative, use only the positive square root, 5. Assume that the conclusion is false and show that Since EG = FG, EG = 5. this assumption leads to a statement that cannot be true. ANSWER: ANSWER: 5 Assume that the conclusion is false and show that Find each measure. this assumption leads to a statement that cannot be 12. RS true.

10. Explain how to locate the largest angle in a scalene triangle. Then explain when a triangle does not have one largest angle. SOLUTION: The largest angle in a scalene triangle is opposite the SOLUTION: longest side. In an isosceles triangle, there may be Here RT = TS. By the converse of the Perpendicular two congruent angles that are larger than the third Bisector Theorem, is a perpendicular bisector of angle, so the sides opposite the congruent angles are

longer than the base. In an equilateral triangle, all angles are the same size. Therefore, .

ANSWER: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third ANSWER: angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all 9 angles are the same size. 13. XZ 11. Find EG if G is the incenter of .

SOLUTION: SOLUTION: By the Incenter Theorem, since G is equidistant from From the figure, the sides of , EG = FG. Find FG using the Thus, Pythagorean Theorem.

Substitute y = 8 in XZ.

Since length cannot be negative, use only the positive ANSWER: square root, 5. 34 Since EG = FG, EG = 5. 14. BASEBALL Jackson, Trevor, and Scott are ANSWER: warming up before a baseball game. One of their 5 warm-up drills requires three players to form a triangle, with one player in the middle. Where should Find each measure. the fourth player stand so that he is the same 12. RS distance from the other three players?

SOLUTION:

Here RT = TS. By the converse of the Perpendicular SOLUTION: Bisector Theorem, is a perpendicular bisector of The players can be represented by the vertices of a

triangle. The point that is equidistant from each Therefore, . vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular bisector of each side of the triangle.

ANSWER: 9

13. XZ

ANSWER:

SOLUTION: From the figure, Thus,

Substitute y = 8 in XZ.

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

ANSWER: The slope of is or So, the slope of 34 the altitude, which is perpendicular to is . 14. BASEBALL Jackson, Trevor, and Scott are warming up before a baseball game. One of their Now, the equation of the altitude from D to is: warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players?

In the same way, we can find the equation of the altitude from E to The slope of is . So, the slope of the

altitude, which is perpendicular to is –2.

SOLUTION: The equation of the altitude is The players can be represented by the vertices of a triangle. The point that is equidistant from each vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular Solve the equations to find the intersection point of bisector of each side of the triangle. the altitudes.

Substitute the value of x in one of the equations to

find the y-coordinate.

ANSWER:

So, the coordinates of the orthocenter of is .

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

The slope of is or So, the slope of ANSWER: (2, 3) the altitude, which is perpendicular to is . 16. PROM Georgia is on the prom committee. She Now, the equation of the altitude from D to is: wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates of the point where the chain should attach to the In the same way, we can find the equation of the triangle? altitude from E to SOLUTION: The slope of is . So, the slope of the In order for the triangles to hang so that they are balanced parallel to the floor, each triangle must be altitude, which is perpendicular to is –2. attached to its chain at its centroid. This point is located at the intersection of the medians of the The equation of the altitude is triangle.

The midpoint of the side from (0, 4) to (6, 0) is Solve the equations to find the intersection point of or (3, 2). The midpoint of the side the altitudes.

from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has Substitute the value of x in one of the equations to endpoints at (0, 4) and (4.5, 4). An equation of the find the y-coordinate. line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4).

So, the coordinates of the orthocenter of is .

ANSWER: (3, 4) List the angles and sides of each triangle in order from smallest to largest. ANSWER: (2, 3)

16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with 17. coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates SOLUTION: of the point where the chain should attach to the The sides from shortest to longest are . triangle? The angles opposite these sides are ∠S, ∠R, and SOLUTION: ∠T, respectively. So the angles from smallest to In order for the triangles to hang so that they are largest are ∠S, ∠R, and ∠T. balanced parallel to the floor, each triangle must be attached to its chain at its centroid. This point is ANSWER: located at the intersection of the medians of the triangle.

The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4). 18.

One median of this triangle has endpoints at (3, 8) SOLUTION: and (3, 2). An equation of the line containing this Use the Triangle Angle-Sum Theorem to find the median is x = 3. Another median of this triangle has angle measures of each angle in the triangle. endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4).

Replace x with 5.6 to find angle measures.

ANSWER: (3, 4) List the angles and sides of each triangle in

order from smallest to largest.

The angles from smallest to largest are ∠N, ∠L, ∠M. The sides opposite these angles are 17. , respectively. So, the sides from shortest to longest are . SOLUTION: The sides from shortest to longest are . ANSWER: The angles opposite these sides are ∠S, ∠R, and ∠N, ∠L, ∠M; , , ∠T, respectively. So the angles from smallest to NEIGHBORHOODS largest are ∠S, ∠R, and ∠T. 19. Anna, Sarah, and Irene live at the intersections of the three roads that make the ANSWER: triangle shown. If the girls want to spend the afternoon together, is it a shorter path for Anna to stop and get Sarah and go on to Irene’s house, or for Sarah to stop and get Irene and then go on to Anna’s house?

18. SOLUTION: Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle.

SOLUTION: The girls' houses can be represented by the vertices of a triangle. List the sides of the triangle in order from shortest to longest. First find the missing angle measure using the Triangle Angle-Sum Theorem.

m∠Irene = 180 – (37 + 53) or 90

Replace x with 5.6 to find angle measures. So, the angles from smallest to largest are ∠Anna, ∠Sarah, ∠Irene. The sides opposite these angles are the path from Sarah to Irene, the path from Irene to Anna, and the path from Sarah to Anna, respectively.

So, the shorter path is for Sarah to get Irene and then go to Anna’s house.

ANSWER: The shorter path is for Sarah to get Irene and then go

to Anna’s house.

State the assumption you would make to start an indirect proof of each statement. The angles from smallest to largest are ∠N, ∠L, 20. ∠M. The sides opposite these angles are SOLUTION: , respectively. So, the sides To start an indirect proof, first assume that what you from shortest to longest are . are trying to prove is false.

ANSWER: ANSWER: ∠N, ∠L, ∠M; , , m∠A < m∠B 19. NEIGHBORHOODS Anna, Sarah, and Irene live at the intersections of the three roads that make the 21. triangle shown. If the girls want to spend the afternoon together, is it a shorter path for Anna to SOLUTION: stop and get Sarah and go on to Irene’s house, or for To start an indirect proof, first assume that what you Sarah to stop and get Irene and then go on to Anna’s are trying to prove is false. is not congruent house? to .

ANSWER: is not congruent to .

22. is a right triangle. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. is not a right triangle. SOLUTION: ANSWER: The girls' houses can be represented by the vertices is not a right triangle. of a triangle. List the sides of the triangle in order from shortest to longest. First find the missing angle 23. If 3y < 12, then y < 4. measure using the Triangle Angle-Sum Theorem. SOLUTION: m∠Irene = 180 – (37 + 53) or 90 To start an indirect proof, first assume that what you are trying to prove is false. If 3y < 12, then y ≥ 4. So, the angles from smallest to largest are ∠Anna, ANSWER: ∠Sarah, ∠Irene. The sides opposite these angles are the path from Sarah to Irene, the path from Irene to Anna, and the path from Sarah to Anna, respectively. 24. Write an indirect proof to show that if two angles are So, the shorter path is for Sarah to get Irene and then complementary, neither angle is a right angle. go to Anna’s house. SOLUTION: ANSWER: To start an indirect proof, first assume that what you The shorter path is for Sarah to get Irene and then go are trying to prove is false. In this case, try to find a to Anna’s house. contradiction if you assume that x or y are right angles. State the assumption you would make to start an indirect proof of each statement. Let the measure of one angle be x and the measure 20. of the other angle be y. By the definition of complementary angles, x + y = 90. SOLUTION: To start an indirect proof, first assume that what you Step 1 Assume that the angle with the measure x is a are trying to prove is false. right angle. Then x = 90.

ANSWER: Step 2 Since x = 90, then x + y > 90. This is a m∠A < m∠B contradiction because we know that x + y = 90.

Step 3 Since the assumption that one angle is a right 21. angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle SOLUTION: is a right angle must be true. To start an indirect proof, first assume that what you are trying to prove is false. is not congruent ANSWER: to . Let the measure of one angle be x and the measure ANSWER: of the other angle be y. By the definition of is not congruent to . complementary angles, x + y = 90. Step 1 Assume that the angle with the measure x is a 22. is a right triangle. right angle. Then x = 90. SOLUTION: Step 2 Since x = 90, then x + y > 90. This is a To start an indirect proof, first assume that what you contradiction because we know that x + y = 90. are trying to prove is false. is not a right Step 3 Since the assumption that one angle is a right triangle. angle leads to a contradiction, the assumption must ANSWER: be false. Therefore, the conclusion that neither angle

is not a right triangle. is a right angle must be true. 23. If 3y < 12, then y < 4. 25. CONCESSIONS Isaac purchased two items at the SOLUTION: concession stand at the Houston Dynamo game and To start an indirect proof, first assume that what you spent over $10. Use indirect reasoning to show that are trying to prove is false. If 3y < 12, then y ≥ 4. at least one of the items he purchased was over $5.

ANSWER: SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a 24. Write an indirect proof to show that if two angles are contradiction if you assume that the cost of item x

complementary, neither angle is a right angle. and the cost of item y are less than or equal to $5.

SOLUTION: Let the cost of one item be x, and the cost of the To start an indirect proof, first assume that what you other item be y. are trying to prove is false. In this case, try to find a Given: x + y > 10 contradiction if you assume that x or y are right Prove: x > 5 or y > 5 angles. Indirect Proof: Step 1 Assume that and . Let the measure of one angle be x and the measure of the other angle be y. By the definition of Step 2 If and , then or complementary angles, x + y = 90. . This is a contradiction because we know that x + y > 50. Step 1 Assume that the angle with the measure x is a

right angle. Then x = 90. Step 3 Since the assumption that and

leads to a contradiction of a known fact, the Step 2 Since x = 90, then x + y > 90. This is a assumption must be false. Therefore, the conclusion contradiction because we know that x + y = 90. that x > 5 or y > 5 must be true. Thus, at least one

item had to be over $5. Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must ANSWER: be false. Therefore, the conclusion that neither angle is a right angle must be true. Let the cost of one item be x, and the cost of the other item be y. ANSWER: Given: x + y > 10 Let the measure of one angle be x and the measure Prove: x > 5 or y > 5 of the other angle be y. By the definition of Indirect Proof: complementary angles, x + y = 90. Step 1 Assume that and . Step 1 Assume that the angle with the measure x is a Step 2 If and , then , or right angle. Then x = 90. . This is a contradiction because we Step 2 Since x = 90, then x + y > 90. This is a know that x + y > 10. contradiction because we know that x + y = 90. Step 3 Since the assumption that x ≤ 5 and Step 3 Since the assumption that one angle is a right leads to a contradiction of a known fact, the angle leads to a contradiction, the assumption must assumption must be false. Therefore, the conclusion be false. Therefore, the conclusion that neither angle that x > 5 or y > 5 must be true. Thus, at least one is a right angle must be true. item had to be over $5. CONCESSIONS 25. Isaac purchased two items at the Is it possible to form a triangle with the given concession stand at the Houston Dynamo game and lengths? If not, explain why not. spent over $10. Use indirect reasoning to show that 26. 5, 6, 9 at least one of the items he purchased was over $5. SOLUTION: SOLUTION: Check each inequality. To start an indirect proof, first assume that what you 5 + 6 > 9 are trying to prove is false. In this case, try to find a 5 + 9 > 6 contradiction if you assume that the cost of item x 6 + 9 > 5 and the cost of item y are less than or equal to $5. Since the sum of each pair of side lengths is greater than the third side length, lengths of 5, 6, and 9 units Let the cost of one item be x, and the cost of the will form a triangle. other item be y. Given: x + y > 10 ANSWER: Prove: x > 5 or y > 5 Yes Indirect Proof: Step 1 Assume that and . 27. 3, 4, 8 Step 2 If and , then or SOLUTION: . This is a contradiction because we 3 + 4 < 8

know that x + y > 50. Since the sum of one pair of side lengths is not greater than the third side length, lengths 3, 4, and 8 Step 3 Since the assumption that and units will not form a triangle. leads to a contradiction of a known fact, the assumption must be false. Therefore, the conclusion ANSWER: that x > 5 or y > 5 must be true. Thus, at least one No; 3 + 4 < 8 item had to be over $5. Find the range for the measure of the third side ANSWER: of a triangle given the measure of two sides. Let the cost of one item be x, and the cost of the 28. 5 ft, 7 ft

other item be y. SOLUTION: Given: x + y > 10 Let n represent the length of the third side. Prove: x > 5 or y > 5 Indirect Proof: According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the Step 1 Assume that and . other two sides. Step 2 If and , then , or . This is a contradiction because we If n is the largest side, then n must be less than 5 + 7. Therefore, n < 12. know that x + y > 10. If n is not the largest side, then 7 is the largest and 7 Step 3 Since the assumption that x ≤ 5 and must be less than 5 + n. Therefore, 2 < n. leads to a contradiction of a known fact, the Combining these two inequalities, we get 2 < n < 12. assumption must be false. Therefore, the conclusion that x > 5 or y > 5 must be true. Thus, at least one ANSWER: item had to be over $5. Let x be the length of the third side. 2 ft < x < 12 ft

Is it possible to form a triangle with the given 29. 10.5 cm, 4 cm lengths? If not, explain why not. 26. 5, 6, 9 SOLUTION: Let n represent the length of the third side. SOLUTION:

Check each inequality. According to the Triangle Inequality Theorem, the 5 + 6 > 9 largest side cannot be greater than the sum of the 5 + 9 > 6 other two sides. 6 + 9 > 5

Since the sum of each pair of side lengths is greater than the third side length, lengths of 5, 6, and 9 units If n is the largest side, then n must be less than 10.5

will form a triangle. + 4. Therefore, n < 14.5. If n is not the largest side, then 10.5 is the largest and ANSWER: 10.5 must be less than 4 + n. Therefore, 6.5 < n.

Yes Combining these two inequalities, we get 6.5 < n < 14.5. 27. 3, 4, 8 ANSWER: SOLUTION: Let x be the length of the third side. 6.5 cm < x < 3 + 4 < 8 Since the sum of one pair of side lengths is not 14.5 cm. greater than the third side length, lengths 3, 4, and 8 units will not form a triangle. 30. BIKES Leonard rides his bike to visit Josh. Since High Street is closed, he has to travel 2 miles down ANSWER: Main Street and turn to travel 3 miles farther on 5th No; 3 + 4 < 8 Street. If the three streets form a triangle with Leonard and Josh’s house as two of the vertices, Find the range for the measure of the third side find the range of the possible distance between of a triangle given the measure of two sides. Leonard and Josh’s houses when traveling straight 28. 5 ft, 7 ft down High Street. SOLUTION: SOLUTION: Let n represent the length of the third side. Let x be the distance between Leonard and Josh’s houses when traveling straight down High Street.

According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the Next, set up and solve each of the three triangle

other two sides. inequalities.

2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1. If n is the largest side, then n must be less than 5 +

7. Therefore, n < 12. Notice that x > –1 is always true for any whole If n is not the largest side, then 7 is the largest and 7 number measure for x. Combining the two remaining must be less than 5 + n. Therefore, 2 < n. inequalities, the range of values that fit both inequalities is x > 1 and x < 5, which can be written Combining these two inequalities, we get 2 < n < 12. as 1 mile < x < 5 miles. Therefore, the distance is

ANSWER: greater than 1 mile and less than 5 miles. Let x be the length of the third side. 2 ft < x < 12 ft ANSWER: The distance is greater than 1 mile and less than 5 29. 10.5 cm, 4 cm miles. SOLUTION: Compare the given measures. Let n represent the length of the third side. 31. m∠ABC, m∠DEF

According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the other two sides.

Combining these two inequalities, we get 6.5 < n < In and , and 14.5. AC > DF. By the Converse of the Hinge Theorem,

ANSWER: ANSWER: Let x be the length of the third side. 6.5 cm < x <

14.5 cm. m∠ABC > m∠DEF

30. BIKES Leonard rides his bike to visit Josh. Since 32. QT and RS High Street is closed, he has to travel 2 miles down Main Street and turn to travel 3 miles farther on 5th Street. If the three streets form a triangle with Leonard and Josh’s house as two of the vertices, find the range of the possible distance between Leonard and Josh’s houses when traveling straight down High Street. SOLUTION: Let x be the distance between Leonard and Josh’s houses when traveling straight down High Street.

Next, set up and solve each of the three triangle SOLUTION: inequalities. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 In and , That is, 5 > x, x > 1, and x > –1. and . By the Hinge Theorem, . Notice that x > –1 is always true for any whole number measure for x. Combining the two remaining ANSWER: inequalities, the range of values that fit both QT > RS inequalities is x > 1 and x < 5, which can be written as 1 mile < x < 5 miles. Therefore, the distance is 33. BOATING Rose and Connor each row across a greater than 1 mile and less than 5 miles. pond heading to the same point. Neither of them has ANSWER: rowed a boat before, so they both go off course as shown in the diagram. After two minutes, they have The distance is greater than 1 mile and less than 5 each traveled 50 yards. Who is closer to their miles. destination?

Compare the given measures. 31. m∠ABC, m∠DEF

SOLUTION: SOLUTION: In and , and As indicated, the distance from the anchor icon to AC > DF. By the Converse of the Hinge Theorem, each boat is congruent and the distanced from the anchor to the destination point (the picnic table icon) ANSWER: is also congruent. We know that Connor's angle is larger than Rose's so, based on the Hinge Theorem, m∠ABC > m∠DEF the distance that Connor has to travel to get to their destination point is further than Rose's. Therefore, Rose is closer to the destination. 32. QT and RS ANSWER: Rose

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: QT > RS

33. BOATING Rose and Connor each row across a pond heading to the same point. Neither of them has rowed a boat before, so they both go off course as shown in the diagram. After two minutes, they have each traveled 50 yards. Who is closer to their destination?

SOLUTION: As indicated, the distance from the anchor icon to each boat is congruent and the distanced from the anchor to the destination point (the picnic table icon) is also congruent. We know that Connor's angle is larger than Rose's so, based on the Hinge Theorem, the distance that Connor has to travel to get to their destination point is further than Rose's. Therefore, Rose is closer to the destination.

ANSWER: Rose State whether each sentence is true or false . If false, replace the underlined term to make a true sentence. 1. The altitudes of a triangle intersect at the centroid. SOLUTION: The centroid is the the point where the medians intersect. The orthocenter is the point where the altitudes intersect. false; orthocenter

ANSWER: false; orthocenter

altitudes intersect. false; orthocenter concurrent lines. The statement is true.

ANSWER: ANSWER: false; orthocenter true

2. The point of concurrency of the medians of a triangle 6. A proof by contradiction uses indirect reasoning.

is called the incenter. SOLUTION: SOLUTION: Indirect reasoning is key when writing a proof by The point where the medians intersect is the contradiction. The statement is true. centroid. The point of concurrency of the angle bisectors of a triangleis called the incenter. The ANSWER: sentence is false. "The point of concurrency of the true angle bisectors of a triangle is called the incenter." is the true sentence. 7. A median of a triangle connects the midpoint of one side of the triangle to the midpoint of another side of ANSWER: the triangle. false; angle bisectors SOLUTION: 3. The circumcenter of a triangle is equidistant from the A median of a triangle connects the vertex to the vertices of the triangle. midpoint of the side opposite it. The sentence is false. The true sentence is "A median of a triangle SOLUTION: connects the midpoint of one side of the triangle to The point that is equidistant from the vertices of a the vertex opposite that side." triangle is called the circumcenter. The statement is true. ANSWER: false; the vertex opposite that side ANSWER: true 8. The incenter is the point at which the angle bisectors of a triangle intersect. 4. To find the centroid of a triangle, first construct the SOLUTION: angle bisectors. The point where the angle bisectors intersect is SOLUTION: called the incenter. The statement is true. To find the centroid of a triangle, first construct the medians. The sentence is false. The true sentence is ANSWER: "To find the centroid of a triangle, first construct the true medians." 9. Explain how to write a proof by contradiction. ANSWER: SOLUTION: false; medians Assume that the conclusion is false and show that 5. The perpendicular bisectors of a triangle are this assumption leads to a statement that cannot be concurrent lines. true. SOLUTION: ANSWER: The perpendicular bisectors of a triangle are Assume that the conclusion is false and show that concurrent lines. The statement is true. this assumption leads to a statement that cannot be true. ANSWER: true 10. Explain how to locate the largest angle in a scalene triangle. Then explain when a triangle does not have 6. A proof by contradiction uses indirect reasoning. one largest angle. SOLUTION: SOLUTION: Indirect reasoning is key when writing a proof by The largest angle in a scalene triangle is opposite the contradiction. The statement is true. longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third ANSWER: angle, so the sides opposite the congruent angles are true longer than the base. In an equilateral triangle, all angles are the same size. 7. A median of a triangle connects the midpoint of one side of the triangle to the midpoint of another side of ANSWER: the triangle. The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be SOLUTION: two congruent angles that are larger than the third A median of a triangle connects the vertex to the angle, so the sides opposite the congruent angles are midpoint of the side opposite it. The sentence is false. longer than the base. In an equilateral triangle, all The true sentence is "A median of a triangle angles are the same size. connects the midpoint of one side of the triangle to the vertex opposite that side." 11. Find EG if G is the incenter of . ANSWER: false; the vertex opposite that side 8. The incenter is the point at which the angle bisectors of a triangle intersect. SOLUTION: SOLUTION: By the Incenter Theorem, since G is equidistant from The point where the angle bisectors intersect is the sides of , EG = FG. Find FG using the called the incenter. The statement is true. Pythagorean Theorem. ANSWER: true 9. Explain how to write a proof by contradiction. SOLUTION: Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true. Since length cannot be negative, use only the positive square root, 5. ANSWER: Since EG = FG, EG = 5. Assume that the conclusion is false and show that Chapterthis assumption 5 Study Guide leads and to a Reviewstatement that cannot be ANSWER: true. 5 10. Explain how to locate the largest angle in a scalene Find each measure. triangle. Then explain when a triangle does not have 12. RS one largest angle. SOLUTION: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are SOLUTION: longer than the base. In an equilateral triangle, all angles are the same size. Here RT = TS. By the converse of the Perpendicular Bisector Theorem, is a perpendicular bisector of ANSWER: The largest angle in a scalene triangle is opposite the Therefore, . longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all angles are the same size. ANSWER: 11. Find EG if G is the incenter of . 9

13. XZ

SOLUTION: By the Incenter Theorem, since G is equidistant from

the sides of , EG = FG. Find FG using the Pythagorean Theorem. SOLUTION: From the figure, Thus,

Substitute y = 8 in XZ.

Since length cannot be negative, use only the positive square root, 5. Since EG = FG, EG = 5.

ANSWER: ANSWER: 5 34 Find each measure. 14. BASEBALL Jackson, Trevor, and Scott are 12. RS warming up before a baseball game. One of their warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players?

eSolutions Manual - Powered by Cognero Page 2 SOLUTION: Here RT = TS. By the converse of the Perpendicular Bisector Theorem, is a perpendicular bisector of

Therefore, . SOLUTION:

The players can be represented by the vertices of a triangle. The point that is equidistant from each vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular ANSWER: bisector of each side of the triangle. 9

13. XZ

SOLUTION: ANSWER: From the figure, Thus,

Substitute y = 8 in XZ.

15. The vertices of are D(0, 0), E(0, 7), and F(6, ANSWER: 3). Find the coordinates of the orthocenter of . 34 SOLUTION: 14. BASEBALL Jackson, Trevor, and Scott are warming up before a baseball game. One of their The slope of is or So, the slope of warm-up drills requires three players to form a triangle, with one player in the middle. Where should the altitude, which is perpendicular to is . the fourth player stand so that he is the same distance from the other three players? Now, the equation of the altitude from D to is:

In the same way, we can find the equation of the

altitude from E to SOLUTION: The slope of is . So, the slope of the The players can be represented by the vertices of a triangle. The point that is equidistant from each altitude, which is perpendicular to is –2. vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular The equation of the altitude is bisector of each side of the triangle.

Solve the equations to find the intersection point of the altitudes.

ANSWER: Substitute the value of x in one of the equations to find the y-coordinate.

So, the coordinates of the orthocenter of is .

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

The slope of is or So, the slope of

the altitude, which is perpendicular to is . Now, the equation of the altitude from D to is: ANSWER: (2, 3)

16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with In the same way, we can find the equation of the coordinates (0, 4), (3, 8), and (6, 0). If each triangle altitude from E to is to be hung by one chain, what are the coordinates The slope of is . So, the slope of the of the point where the chain should attach to the triangle? altitude, which is perpendicular to is –2. SOLUTION: The equation of the altitude is In order for the triangles to hang so that they are balanced parallel to the floor, each triangle must be attached to its chain at its centroid. This point is located at the intersection of the medians of the Solve the equations to find the intersection point of triangle. the altitudes. The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4).

Substitute the value of x in one of the equations to find the y-coordinate. One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s So, the coordinates of the orthocenter of is centroid, is the point (3, 4). .

ANSWER: (3, 4) ANSWER: (2, 3) List the angles and sides of each triangle in order from smallest to largest. 16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates of the point where the chain should attach to the triangle? 17. SOLUTION: SOLUTION: In order for the triangles to hang so that they are The sides from shortest to longest are . balanced parallel to the floor, each triangle must be The angles opposite these sides are ∠S, ∠R, and attached to its chain at its centroid. This point is ∠T, respectively. So the angles from smallest to located at the intersection of the medians of the largest are ∠S, ∠R, and ∠T. triangle. ANSWER: The midpoint of the side from (0, 4) to (6, 0) is

or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has 18. endpoints at (0, 4) and (4.5, 4). An equation of the SOLUTION: line containing this median is y = 4. The intersection Use the Triangle Angle-Sum Theorem to find the of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4). angle measures of each angle in the triangle.

Replace x with 5.6 to find angle measures. ANSWER: (3, 4)

List the angles and sides of each triangle in order from smallest to largest.

17. The angles from smallest to largest are N, L, SOLUTION: ∠ ∠ The sides from shortest to longest are . ∠M. The sides opposite these angles are , respectively. So, the sides The angles opposite these sides are S, R, and ∠ ∠ from shortest to longest are . ∠T, respectively. So the angles from smallest to largest are ∠S, ∠R, and ∠T. ANSWER: ∠N, ∠L, ∠M; , , ANSWER: 19. NEIGHBORHOODS Anna, Sarah, and Irene live at the intersections of the three roads that make the triangle shown. If the girls want to spend the afternoon together, is it a shorter path for Anna to stop and get Sarah and go on to Irene’s house, or for Sarah to stop and get Irene and then go on to Anna’s house?

18. SOLUTION: Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle.

So, the angles from smallest to largest are ∠Anna, ∠Sarah, ∠Irene. The sides opposite these angles are the path from Sarah to Irene, the path from Irene to Anna, and the path from Sarah to Anna, respectively. So, the shorter path is for Sarah to get Irene and then

go to Anna’s house. ANSWER: The shorter path is for Sarah to get Irene and then go to Anna’s house. The angles from smallest to largest are ∠N, ∠L, ∠M. The sides opposite these angles are State the assumption you would make to start , respectively. So, the sides an indirect proof of each statement. from shortest to longest are . 20.

ANSWER: SOLUTION: To start an indirect proof, first assume that what you ∠N, ∠L, ∠M; , , are trying to prove is false. 19. NEIGHBORHOODS Anna, Sarah, and Irene live ANSWER: at the intersections of the three roads that make the triangle shown. If the girls want to spend the m∠A < m∠B afternoon together, is it a shorter path for Anna to stop and get Sarah and go on to Irene’s house, or for 21. Sarah to stop and get Irene and then go on to Anna’s house? SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. is not congruent to .

ANSWER: is not congruent to .

22. is a right triangle. SOLUTION: SOLUTION: To start an indirect proof, first assume that what you The girls' houses can be represented by the vertices are trying to prove is false. is not a right of a triangle. List the sides of the triangle in order triangle. from shortest to longest. First find the missing angle ANSWER: measure using the Triangle Angle-Sum Theorem. is not a right triangle. m∠Irene = 180 – (37 + 53) or 90 23. If 3y < 12, then y < 4. So, the angles from smallest to largest are ∠Anna, SOLUTION: ∠Sarah, ∠Irene. The sides opposite these angles are To start an indirect proof, first assume that what you the path from Sarah to Irene, the path from Irene to are trying to prove is false. If 3y < 12, then y ≥ 4. Anna, and the path from Sarah to Anna, respectively. So, the shorter path is for Sarah to get Irene and then ANSWER: go to Anna’s house.

ANSWER: 24. Write an indirect proof to show that if two angles are The shorter path is for Sarah to get Irene and then go complementary, neither angle is a right angle. to Anna’s house. SOLUTION: To start an indirect proof, first assume that what you State the assumption you would make to start are trying to prove is false. In this case, try to find a an indirect proof of each statement. contradiction if you assume that x or y are right 20. angles. SOLUTION: Let the measure of one angle be x and the measure To start an indirect proof, first assume that what you of the other angle be y. By the definition of

are trying to prove is false. complementary angles, x + y = 90. ANSWER: Step 1 Assume that the angle with the measure x is a m∠A < m∠B right angle. Then x = 90.

21. Step 2 Since x = 90, then x + y > 90. This is a contradiction because we know that x + y = 90. SOLUTION: To start an indirect proof, first assume that what you Step 3 Since the assumption that one angle is a right are trying to prove is false. is not congruent angle leads to a contradiction, the assumption must to . be false. Therefore, the conclusion that neither angle is a right angle must be true. ANSWER: ANSWER: is not congruent to . Let the measure of one angle be x and the measure 22. is a right triangle. of the other angle be y. By the definition of

SOLUTION: complementary angles, x + y = 90. To start an indirect proof, first assume that what you Step 1 Assume that the angle with the measure x is a are trying to prove is false. is not a right right angle. Then x = 90. triangle. Step 2 Since x = 90, then x + y > 90. This is a ANSWER: contradiction because we know that x + y = 90. is not a right triangle. Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must 23. If 3y < 12, then y < 4. be false. Therefore, the conclusion that neither angle SOLUTION: is a right angle must be true. To start an indirect proof, first assume that what you are trying to prove is false. If 3y < 12, then y ≥ 4. 25. CONCESSIONS Isaac purchased two items at the ANSWER: concession stand at the Houston Dynamo game and spent over $10. Use indirect reasoning to show that at least one of the items he purchased was over $5. 24. Write an indirect proof to show that if two angles are SOLUTION: complementary, neither angle is a right angle. To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a SOLUTION: contradiction if you assume that the cost of item x To start an indirect proof, first assume that what you and the cost of item y are less than or equal to $5. are trying to prove is false. In this case, try to find a contradiction if you assume that x or y are right Let the cost of one item be x, and the cost of the angles. other item be y.

Given: x + y > 10 Let the measure of one angle be x and the measure Prove: x > 5 or y > 5 of the other angle be y. By the definition of Indirect Proof: complementary angles, x + y = 90. Step 1 Assume that and .

Step 1 Assume that the angle with the measure x is a right angle. Then x = 90. Step 2 If and , then or . This is a contradiction because we Step 2 Since x = 90, then x + y > 90. This is a know that x + y > 50. contradiction because we know that x + y = 90. Step 3 Since the assumption that and Step 3 Since the assumption that one angle is a right leads to a contradiction of a known fact, the angle leads to a contradiction, the assumption must assumption must be false. Therefore, the conclusion be false. Therefore, the conclusion that neither angle that x > 5 or y > 5 must be true. Thus, at least one is a right angle must be true. item had to be over $5.

ANSWER: ANSWER: Let the measure of one angle be x and the measure Let the cost of one item be x, and the cost of the of the other angle be y. By the definition of other item be y. complementary angles, x + y = 90. Given: x + y > 10 Step 1 Assume that the angle with the measure x is a Prove: x > 5 or y > 5 right angle. Then x = 90. Indirect Proof: Step 2 Since x = 90, then x + y > 90. This is a Step 1 Assume that and . contradiction because we know that x + y = 90. Step 2 If and , then , or Step 3 Since the assumption that one angle is a right . This is a contradiction because we angle leads to a contradiction, the assumption must know that x + y > 10. be false. Therefore, the conclusion that neither angle Step 3 Since the assumption that x ≤ 5 and is a right angle must be true. leads to a contradiction of a known fact, the assumption must be false. Therefore, the conclusion 25. CONCESSIONS Isaac purchased two items at the concession stand at the Houston Dynamo game and that x > 5 or y > 5 must be true. Thus, at least one spent over $10. Use indirect reasoning to show that item had to be over $5. at least one of the items he purchased was over $5. Is it possible to form a triangle with the given SOLUTION: lengths? If not, explain why not. To start an indirect proof, first assume that what you 26. 5, 6, 9 are trying to prove is false. In this case, try to find a contradiction if you assume that the cost of item x SOLUTION: and the cost of item y are less than or equal to $5. Check each inequality. 5 + 6 > 9 Let the cost of one item be x, and the cost of the 5 + 9 > 6 other item be y. 6 + 9 > 5 Given: x + y > 10 Since the sum of each pair of side lengths is greater Prove: x > 5 or y > 5 than the third side length, lengths of 5, 6, and 9 units Indirect Proof: will form a triangle. Step 1 Assume that and . ANSWER:

Step 2 If and , then or Yes . This is a contradiction because we know that x + y > 50. 27. 3, 4, 8

Step 3 Since the assumption that and SOLUTION: leads to a contradiction of a known fact, the 3 + 4 < 8 assumption must be false. Therefore, the conclusion Since the sum of one pair of side lengths is not that x > 5 or y > 5 must be true. Thus, at least one greater than the third side length, lengths 3, 4, and 8 item had to be over $5. units will not form a triangle.

ANSWER: ANSWER: Let the cost of one item be x, and the cost of the No; 3 + 4 < 8 other item be y. Find the range for the measure of the third side Given: x + y > 10 of a triangle given the measure of two sides. Prove: x > 5 or y > 5 28. 5 ft, 7 ft Indirect Proof: SOLUTION: Step 1 Assume that and . Let n represent the length of the third side.

Step 2 If and , then , or According to the Triangle Inequality Theorem, the . This is a contradiction because we largest side cannot be greater than the sum of the know that x + y > 10. other two sides.

Step 3 Since the assumption that x ≤ 5 and If n is the largest side, then n must be less than 5 + leads to a contradiction of a known fact, the 7. Therefore, n < 12. assumption must be false. Therefore, the conclusion If n is not the largest side, then 7 is the largest and 7 must be less than 5 + n. Therefore, 2 < n. that x > 5 or y > 5 must be true. Thus, at least one

item had to be over $5. Combining these two inequalities, we get 2 < n < 12.

Is it possible to form a triangle with the given ANSWER: lengths? If not, explain why not. Let x be the length of the third side. 2 ft < x < 12 ft 26. 5, 6, 9 SOLUTION: 29. 10.5 cm, 4 cm Check each inequality. 5 + 6 > 9 SOLUTION: 5 + 9 > 6 Let n represent the length of the third side. 6 + 9 > 5 Since the sum of each pair of side lengths is greater According to the Triangle Inequality Theorem, the than the third side length, lengths of 5, 6, and 9 units largest side cannot be greater than the sum of the will form a triangle. other two sides.

ANSWER: If n is the largest side, then n must be less than 10.5 Yes + 4. Therefore, n < 14.5. If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n. 27. 3, 4, 8 SOLUTION: Combining these two inequalities, we get 6.5 < n < 3 + 4 < 8 14.5. Since the sum of one pair of side lengths is not ANSWER: greater than the third side length, lengths 3, 4, and 8 units will not form a triangle. Let x be the length of the third side. 6.5 cm < x < 14.5 cm. ANSWER: No; 3 + 4 < 8 30. BIKES Leonard rides his bike to visit Josh. Since High Street is closed, he has to travel 2 miles down Find the range for the measure of the third side Main Street and turn to travel 3 miles farther on 5th of a triangle given the measure of two sides. Street. If the three streets form a triangle with 28. 5 ft, 7 ft Leonard and Josh’s house as two of the vertices, SOLUTION: find the range of the possible distance between Leonard and Josh’s houses when traveling straight Let n represent the length of the third side. down High Street.

According to the Triangle Inequality Theorem, the SOLUTION: largest side cannot be greater than the sum of the Let x be the distance between Leonard and Josh’s other two sides. houses when traveling straight down High Street.

If n is the largest side, then n must be less than 5 + Next, set up and solve each of the three triangle 7. Therefore, n < 12. inequalities. If n is not the largest side, then 7 is the largest and 7 2 + 3 > x, 2 + x > 3, and 3 + x > 2 must be less than 5 + n. Therefore, 2 < n. That is, 5 > x, x > 1, and x > –1.

Combining these two inequalities, we get 2 < n < 12. Notice that x > –1 is always true for any whole number measure for x. Combining the two remaining ANSWER: inequalities, the range of values that fit both Let x be the length of the third side. 2 ft < x < 12 ft inequalities is x > 1 and x < 5, which can be written as 1 mile < x < 5 miles. Therefore, the distance is greater than 1 mile and less than 5 miles. 29. 10.5 cm, 4 cm ANSWER: SOLUTION: The distance is greater than 1 mile and less than 5 Let n represent the length of the third side. miles. According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the Compare the given measures. other two sides. 31. m∠ABC, m∠DEF

If n is the largest side, then n must be less than 10.5 + 4. Therefore, n < 14.5. If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n.

Combining these two inequalities, we get 6.5 < n <

14.5. SOLUTION: ANSWER: In and , and Let x be the length of the third side. 6.5 cm < x < AC > DF. By the Converse of the Hinge Theorem, 14.5 cm.

30. BIKES Leonard rides his bike to visit Josh. Since ANSWER: High Street is closed, he has to travel 2 miles down m∠ABC > m∠DEF Main Street and turn to travel 3 miles farther on 5th Street. If the three streets form a triangle with Leonard and Josh’s house as two of the vertices, 32. QT and RS find the range of the possible distance between Leonard and Josh’s houses when traveling straight down High Street. SOLUTION: Let x be the distance between Leonard and Josh’s houses when traveling straight down High Street.

Next, set up and solve each of the three triangle inequalities. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1. SOLUTION: Notice that x > –1 is always true for any whole In and , number measure for x. Combining the two remaining and . inequalities, the range of values that fit both inequalities is x > 1 and x < 5, which can be written By the Hinge Theorem, . as 1 mile < x < 5 miles. Therefore, the distance is ANSWER: greater than 1 mile and less than 5 miles. QT > RS ANSWER: The distance is greater than 1 mile and less than 5 33. BOATING Rose and Connor each row across a pond heading to the same point. Neither of them has

miles. rowed a boat before, so they both go off course as shown in the diagram. After two minutes, they have Compare the given measures. each traveled 50 yards. Who is closer to their 31. m∠ABC, m∠DEF destination?

SOLUTION: In and , and AC > DF. By the Converse of the Hinge Theorem,

SOLUTION: ANSWER: As indicated, the distance from the anchor icon to m∠ABC > m∠DEF each boat is congruent and the distanced from the anchor to the destination point (the picnic table icon) is also congruent. We know that Connor's angle is 32. QT and RS larger than Rose's so, based on the Hinge Theorem, the distance that Connor has to travel to get to their destination point is further than Rose's. Therefore, Rose is closer to the destination.

ANSWER: Rose

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: QT > RS

ANSWER: false; orthocenter

ANSWER: false; medians 5. The perpendicular bisectors of a triangle are concurrent lines. State whether each sentence is true or false . If SOLUTION: false, replace the underlined term to make a The perpendicular bisectors of a triangle are true sentence. concurrent lines. The statement is true. 1. The altitudes of a triangle intersect at the centroid. ANSWER: SOLUTION: true The centroid is the the point where the medians intersect. The orthocenter is the point where the 6. A proof by contradiction uses indirect reasoning. altitudes intersect. false; orthocenter SOLUTION: ANSWER: Indirect reasoning is key when writing a proof by false; orthocenter contradiction. The statement is true. ANSWER: 2. The point of concurrency of the medians of a triangle is called the incenter. true SOLUTION: 7. A median of a triangle connects the midpoint of one The point where the medians intersect is the side of the triangle to the midpoint of another side of centroid. The point of concurrency of the angle the triangle. bisectors of a triangleis called the incenter. The SOLUTION: sentence is false. "The point of concurrency of the A median of a triangle connects the vertex to the angle bisectors of a triangle is called the incenter." is midpoint of the side opposite it. The sentence is false.

the true sentence. The true sentence is "A median of a triangle ANSWER: connects the midpoint of one side of the triangle to the vertex opposite that side." false; angle bisectors ANSWER: 3. The circumcenter of a triangle is equidistant from the vertices of the triangle. false; the vertex opposite that side SOLUTION: 8. The incenter is the point at which the angle bisectors The point that is equidistant from the vertices of a of a triangle intersect. triangle is called the circumcenter. The statement is SOLUTION:

true. The point where the angle bisectors intersect is ANSWER: called the incenter. The statement is true. true ANSWER: 4. To find the centroid of a triangle, first construct the true

angle bisectors. 9. Explain how to write a proof by contradiction. SOLUTION: SOLUTION: To find the centroid of a triangle, first construct the Assume that the conclusion is false and show that medians. The sentence is false. The true sentence is this assumption leads to a statement that cannot be "To find the centroid of a triangle, first construct the true. medians." ANSWER: ANSWER: Assume that the conclusion is false and show that false; medians this assumption leads to a statement that cannot be 5. The perpendicular bisectors of a triangle are true.

concurrent lines. 10. Explain how to locate the largest angle in a scalene SOLUTION: triangle. Then explain when a triangle does not have The perpendicular bisectors of a triangle are one largest angle. concurrent lines. The statement is true. SOLUTION: ANSWER: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be true two congruent angles that are larger than the third 6. A proof by contradiction uses indirect reasoning. angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all SOLUTION: angles are the same size. Indirect reasoning is key when writing a proof by contradiction. The statement is true. ANSWER: The largest angle in a scalene triangle is opposite the ANSWER: longest side. In an isosceles triangle, there may be true two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are 7. A median of a triangle connects the midpoint of one longer than the base. In an equilateral triangle, all side of the triangle to the midpoint of another side of angles are the same size. the triangle. 11. Find EG if G is the incenter of . SOLUTION: A median of a triangle connects the vertex to the midpoint of the side opposite it. The sentence is false. The true sentence is "A median of a triangle connects the midpoint of one side of the triangle to the vertex opposite that side." SOLUTION: ANSWER: By the Incenter Theorem, since G is equidistant from false; the vertex opposite that side the sides of , EG = FG. Find FG using the Pythagorean Theorem. 8. The incenter is the point at which the angle bisectors of a triangle intersect. SOLUTION: The point where the angle bisectors intersect is called the incenter. The statement is true.

ANSWER:

true Since length cannot be negative, use only the positive 9. Explain how to write a proof by contradiction. square root, 5. Since EG = FG, EG = 5. SOLUTION: Assume that the conclusion is false and show that ANSWER: this assumption leads to a statement that cannot be 5 true. Find each measure. ANSWER: 12. RS Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true.

10. Explain how to locate the largest angle in a scalene triangle. Then explain when a triangle does not have one largest angle. SOLUTION: SOLUTION: Here RT = TS. By the converse of the Perpendicular The largest angle in a scalene triangle is opposite the Bisector Theorem, is a perpendicular bisector of longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third Therefore, . angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all angles are the same size.

ANSWER: The largest angle in a scalene triangle is opposite the ANSWER: longest side. In an isosceles triangle, there may be 9 two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all 13. XZ angles are the same size.

11. Find EG if G is the incenter of .

SOLUTION: From the figure, SOLUTION: Thus, By the Incenter Theorem, since G is equidistant from the sides of , EG = FG. Find FG using the Pythagorean Theorem. Substitute y = 8 in XZ.

ANSWER: 34 Since length cannot be negative, use only the positive square root, 5. 14. BASEBALL Jackson, Trevor, and Scott are Since EG = FG, EG = 5. warming up before a baseball game. One of their warm-up drills requires three players to form a ANSWER: triangle, with one player in the middle. Where should 5 the fourth player stand so that he is the same distance from the other three players? Find each measure. 12. RS

SOLUTION: SOLUTION: The players can be represented by the vertices of a Here RT = TS. By the converse of the Perpendicular triangle. The point that is equidistant from each Bisector Theorem, is a perpendicular bisector of vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular Therefore, . bisector of each side of the triangle.

ANSWER: 9

13. XZ ANSWER:

SOLUTION: From the figure, Thus,

Substitute y = 8 in XZ. 15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

The slope of is or So, the slope of

ChapterANSWER: 5 Study Guide and Review the altitude, which is perpendicular to is . 34 Now, the equation of the altitude from D to is: 14. BASEBALL Jackson, Trevor, and Scott are warming up before a baseball game. One of their warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players? In the same way, we can find the equation of the altitude from E to The slope of is . So, the slope of the altitude, which is perpendicular to is –2.

The equation of the altitude is

SOLUTION: The players can be represented by the vertices of a Solve the equations to find the intersection point of triangle. The point that is equidistant from each the altitudes. vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular bisector of each side of the triangle.

Substitute the value of x in one of the equations to find the y-coordinate.

ANSWER: So, the coordinates of the orthocenter of is .

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION: ANSWER: (2, 3) The slope of is or So, the slope of 16. PROM Georgia is on the prom committee. She the altitude, which is perpendicular to is . wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She Now, the equation of the altitude from D to is: sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates eSolutions Manual - Powered by Cognero of the point where the chain should attach to thePage 3 triangle? SOLUTION: In the same way, we can find the equation of the In order for the triangles to hang so that they are altitude from E to balanced parallel to the floor, each triangle must be attached to its chain at its centroid. This point is The slope of is . So, the slope of the located at the intersection of the medians of the altitude, which is perpendicular to is –2. triangle.

The equation of the altitude is The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

Solve the equations to find the intersection point of the altitudes. from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection Substitute the value of x in one of the equations to of x = 3 and y = 4, and the location of the traingle’s find the y-coordinate. centroid, is the point (3, 4).

So, the coordinates of the orthocenter of is .

ANSWER: (3, 4) List the angles and sides of each triangle in order from smallest to largest.

ANSWER: (2, 3)

16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the 17. ceiling so that they are parallel to the floor. She SOLUTION: sketched out one triangle on a coordinate plane with The sides from shortest to longest are . coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates The angles opposite these sides are ∠S, ∠R, and of the point where the chain should attach to the ∠T, respectively. So the angles from smallest to triangle? largest are ∠S, ∠R, and ∠T. SOLUTION: ANSWER: In order for the triangles to hang so that they are balanced parallel to the floor, each triangle must be attached to its chain at its centroid. This point is located at the intersection of the medians of the triangle.

The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side 18.

from (3, 8) to (6, 0) is or (4.5, 4). SOLUTION: Use the Triangle Angle-Sum Theorem to find the One median of this triangle has endpoints at (3, 8) angle measures of each angle in the triangle. and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4).

Replace x with 5.6 to find angle measures.

ANSWER: (3, 4)

List the angles and sides of each triangle in order from smallest to largest.

The angles from smallest to largest are ∠N, ∠L, ∠M. The sides opposite these angles are , respectively. So, the sides from shortest to longest are . 17. ANSWER: SOLUTION: ∠N, ∠L, ∠M; , , The sides from shortest to longest are . The angles opposite these sides are ∠S, ∠R, and 19. NEIGHBORHOODS Anna, Sarah, and Irene live at the intersections of the three roads that make the T, respectively. So the angles from smallest to ∠ triangle shown. If the girls want to spend the largest are ∠S, ∠R, and ∠T. afternoon together, is it a shorter path for Anna to stop and get Sarah and go on to Irene’s house, or for ANSWER: Sarah to stop and get Irene and then go on to Anna’s house?

18. SOLUTION:

Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle. SOLUTION: The girls' houses can be represented by the vertices of a triangle. List the sides of the triangle in order from shortest to longest. First find the missing angle measure using the Triangle Angle-Sum Theorem.

m∠Irene = 180 – (37 + 53) or 90

So, the angles from smallest to largest are ∠Anna, ∠Sarah, ∠Irene. The sides opposite these angles are Replace x with 5.6 to find angle measures. the path from Sarah to Irene, the path from Irene to Anna, and the path from Sarah to Anna, respectively. So, the shorter path is for Sarah to get Irene and then go to Anna’s house.

ANSWER: The shorter path is for Sarah to get Irene and then go to Anna’s house.

State the assumption you would make to start an indirect proof of each statement. 20. SOLUTION: The angles from smallest to largest are ∠N, ∠L, To start an indirect proof, first assume that what you ∠M. The sides opposite these angles are are trying to prove is false. , respectively. So, the sides from shortest to longest are . ANSWER: m∠A < m∠B ANSWER: ∠N, ∠L, ∠M; , , 21. 19. NEIGHBORHOODS Anna, Sarah, and Irene live SOLUTION: at the intersections of the three roads that make the To start an indirect proof, first assume that what you triangle shown. If the girls want to spend the are trying to prove is false. is not congruent afternoon together, is it a shorter path for Anna to

stop and get Sarah and go on to Irene’s house, or for to . Sarah to stop and get Irene and then go on to Anna’s ANSWER: house? is not congruent to .

22. is a right triangle. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. is not a right triangle.

ANSWER:

is not a right triangle. SOLUTION: The girls' houses can be represented by the vertices 23. If 3y < 12, then y < 4. of a triangle. List the sides of the triangle in order SOLUTION: from shortest to longest. First find the missing angle measure using the Triangle Angle-Sum Theorem. To start an indirect proof, first assume that what you are trying to prove is false. If 3y < 12, then y ≥ 4. m∠Irene = 180 – (37 + 53) or 90 ANSWER:

to Anna’s house. Let the measure of one angle be x and the measure of the other angle be y. By the definition of State the assumption you would make to start complementary angles, x + y = 90. an indirect proof of each statement.

20. Step 1 Assume that the angle with the measure x is a SOLUTION: right angle. Then x = 90.

To start an indirect proof, first assume that what you Step 2 Since x = 90, then x + y > 90. This is a are trying to prove is false. contradiction because we know that x + y = 90. ANSWER: Step 3 Since the assumption that one angle is a right m∠A < m∠B angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle 21. is a right angle must be true. SOLUTION: ANSWER: To start an indirect proof, first assume that what you Let the measure of one angle be x and the measure are trying to prove is false. is not congruent of the other angle be y. By the definition of to . complementary angles, x + y = 90. ANSWER: Step 1 Assume that the angle with the measure x is a is not congruent to . right angle. Then x = 90. Step 2 Since x = 90, then x + y > 90. This is a 22. is a right triangle. contradiction because we know that x + y = 90. SOLUTION: Step 3 Since the assumption that one angle is a right To start an indirect proof, first assume that what you angle leads to a contradiction, the assumption must are trying to prove is false. is not a right triangle. be false. Therefore, the conclusion that neither angle is a right angle must be true. ANSWER: is not a right triangle. 25. CONCESSIONS Isaac purchased two items at the concession stand at the Houston Dynamo game and 23. If 3y < 12, then y < 4. spent over $10. Use indirect reasoning to show that

SOLUTION: at least one of the items he purchased was over $5. To start an indirect proof, first assume that what you SOLUTION: are trying to prove is false. If 3y < 12, then y ≥ 4. To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a ANSWER: contradiction if you assume that the cost of item x and the cost of item y are less than or equal to $5.

24. Write an indirect proof to show that if two angles are Let the cost of one item be x, and the cost of the complementary, neither angle is a right angle. other item be y. SOLUTION: Given: x + y > 10 To start an indirect proof, first assume that what you Prove: x > 5 or y > 5 are trying to prove is false. In this case, try to find a Indirect Proof: contradiction if you assume that x or y are right Step 1 Assume that and . angles. Step 2 If and , then or Let the measure of one angle be x and the measure . This is a contradiction because we of the other angle be y. By the definition of know that x + y > 50. complementary angles, x + y = 90. Step 3 Since the assumption that and Step 1 Assume that the angle with the measure x is a leads to a contradiction of a known fact, the right angle. Then x = 90. assumption must be false. Therefore, the conclusion that x > 5 or y > 5 must be true. Thus, at least one Step 2 Since x = 90, then x + y > 90. This is a item had to be over $5. contradiction because we know that x + y = 90. ANSWER: Step 3 Since the assumption that one angle is a right Let the cost of one item be x, and the cost of the angle leads to a contradiction, the assumption must other item be y. be false. Therefore, the conclusion that neither angle is a right angle must be true. Given: x + y > 10 Prove: x > 5 or y > 5 ANSWER: Indirect Proof: Let the measure of one angle be x and the measure Step 1 Assume that and . of the other angle be y. By the definition of Step 2 If and , then , or complementary angles, x + y = 90. . This is a contradiction because we Step 1 Assume that the angle with the measure x is a know that x + y > 10. right angle. Then x = 90. Step 3 Since the assumption that x ≤ 5 and Step 2 Since x = 90, then x + y > 90. This is a leads to a contradiction of a known fact, the contradiction because we know that x + y = 90. assumption must be false. Therefore, the conclusion Step 3 Since the assumption that one angle is a right that x > 5 or y > 5 must be true. Thus, at least one angle leads to a contradiction, the assumption must item had to be over $5. be false. Therefore, the conclusion that neither angle is a right angle must be true. Is it possible to form a triangle with the given lengths? If not, explain why not. 25. CONCESSIONS Isaac purchased two items at the 26. 5, 6, 9 concession stand at the Houston Dynamo game and spent over $10. Use indirect reasoning to show that SOLUTION: at least one of the items he purchased was over $5. Check each inequality. 5 + 6 > 9 SOLUTION: 5 + 9 > 6 To start an indirect proof, first assume that what you 6 + 9 > 5 are trying to prove is false. In this case, try to find a Since the sum of each pair of side lengths is greater contradiction if you assume that the cost of item x than the third side length, lengths of 5, 6, and 9 units and the cost of item y are less than or equal to $5. will form a triangle.

Let the cost of one item be x, and the cost of the ANSWER: other item be y. Yes Given: x + y > 10 Prove: x > 5 or y > 5 27. 3, 4, 8 Indirect Proof: Step 1 Assume that and . SOLUTION: 3 + 4 < 8 Step 2 If and , then or Since the sum of one pair of side lengths is not . This is a contradiction because we greater than the third side length, lengths 3, 4, and 8 know that x + y > 50. units will not form a triangle.

Step 3 Since the assumption that and ANSWER: leads to a contradiction of a known fact, the No; 3 + 4 < 8 assumption must be false. Therefore, the conclusion that x > 5 or y > 5 must be true. Thus, at least one Find the range for the measure of the third side item had to be over $5. of a triangle given the measure of two sides. 28. 5 ft, 7 ft ANSWER: SOLUTION: Let the cost of one item be x, and the cost of the Let n represent the length of the third side. other item be y. Given: x + y > 10 According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the Prove: x > 5 or y > 5 other two sides. Indirect Proof: Step 1 Assume that and . If n is the largest side, then n must be less than 5 +

Step 2 If and , then , or 7. Therefore, n < 12. If n is not the largest side, then 7 is the largest and 7 . This is a contradiction because we must be less than 5 + n. Therefore, 2 < n. know that x + y > 10. Step 3 Since the assumption that x ≤ 5 and Combining these two inequalities, we get 2 < n < 12. leads to a contradiction of a known fact, the ANSWER: assumption must be false. Therefore, the conclusion Let x be the length of the third side. 2 ft < x < 12 ft that x > 5 or y > 5 must be true. Thus, at least one item had to be over $5. 29. 10.5 cm, 4 cm

Is it possible to form a triangle with the given SOLUTION: lengths? If not, explain why not. Let n represent the length of the third side. 26. 5, 6, 9 According to the Triangle Inequality Theorem, the SOLUTION: largest side cannot be greater than the sum of the Check each inequality. other two sides. 5 + 6 > 9 5 + 9 > 6 If n is the largest side, then n must be less than 10.5 6 + 9 > 5 + 4. Therefore, n < 14.5. Since the sum of each pair of side lengths is greater If n is not the largest side, then 10.5 is the largest and than the third side length, lengths of 5, 6, and 9 units 10.5 must be less than 4 + n. Therefore, 6.5 < n. will form a triangle. Combining these two inequalities, we get 6.5 < n < ANSWER: 14.5. Yes ANSWER: 27. 3, 4, 8 Let x be the length of the third side. 6.5 cm < x < 14.5 cm. SOLUTION: 3 + 4 < 8 30. BIKES Leonard rides his bike to visit Josh. Since Since the sum of one pair of side lengths is not High Street is closed, he has to travel 2 miles down greater than the third side length, lengths 3, 4, and 8 Main Street and turn to travel 3 miles farther on 5th

units will not form a triangle. Street. If the three streets form a triangle with ANSWER: Leonard and Josh’s house as two of the vertices, find the range of the possible distance between No; 3 + 4 < 8 Leonard and Josh’s houses when traveling straight down High Street. Find the range for the measure of the third side of a triangle given the measure of two sides. SOLUTION: 28. 5 ft, 7 ft Let x be the distance between Leonard and Josh’s houses when traveling straight down High Street. SOLUTION:

Let n represent the length of the third side. Next, set up and solve each of the three triangle inequalities. According to the Triangle Inequality Theorem, the 2 + 3 > x, 2 + x > 3, and 3 + x > 2 largest side cannot be greater than the sum of the That is, 5 > x, x > 1, and x > –1. other two sides. Notice that x > –1 is always true for any whole If n is the largest side, then n must be less than 5 + number measure for x. Combining the two remaining 7. Therefore, n < 12. inequalities, the range of values that fit both If n is not the largest side, then 7 is the largest and 7 inequalities is x > 1 and x < 5, which can be written must be less than 5 + n. Therefore, 2 < n. as 1 mile < x < 5 miles. Therefore, the distance is greater than 1 mile and less than 5 miles. Combining these two inequalities, we get 2 < n < 12. ANSWER: ANSWER: The distance is greater than 1 mile and less than 5 Let x be the length of the third side. 2 ft < x < 12 ft miles.

29. 10.5 cm, 4 cm Compare the given measures. 31. m∠ABC, m∠DEF SOLUTION: Let n represent the length of the third side.

According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the other two sides.

If n is the largest side, then n must be less than 10.5 + 4. Therefore, n < 14.5. SOLUTION: If n is not the largest side, then 10.5 is the largest and In and , and 10.5 must be less than 4 + n. Therefore, 6.5 < n. AC > DF. By the Converse of the Hinge Theorem,

Combining these two inequalities, we get 6.5 < n < 14.5. ANSWER: ANSWER: m∠ABC > m∠DEF Let x be the length of the third side. 6.5 cm < x < 14.5 cm. 32. QT and RS

30. BIKES Leonard rides his bike to visit Josh. Since High Street is closed, he has to travel 2 miles down Main Street and turn to travel 3 miles farther on 5th Street. If the three streets form a triangle with Leonard and Josh’s house as two of the vertices, find the range of the possible distance between Leonard and Josh’s houses when traveling straight down High Street. SOLUTION: Let x be the distance between Leonard and Josh’s houses when traveling straight down High Street. SOLUTION: In and , Next, set up and solve each of the three triangle and . inequalities. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 By the Hinge Theorem, . That is, 5 > x, x > 1, and x > –1. ANSWER:

Notice that x > –1 is always true for any whole QT > RS number measure for x. Combining the two remaining inequalities, the range of values that fit both 33. BOATING Rose and Connor each row across a inequalities is x > 1 and x < 5, which can be written pond heading to the same point. Neither of them has as 1 mile < x < 5 miles. Therefore, the distance is rowed a boat before, so they both go off course as greater than 1 mile and less than 5 miles. shown in the diagram. After two minutes, they have each traveled 50 yards. Who is closer to their ANSWER: destination? The distance is greater than 1 mile and less than 5 miles.

Compare the given measures. 31. m∠ABC, m∠DEF

SOLUTION: As indicated, the distance from the anchor icon to SOLUTION: each boat is congruent and the distanced from the In and , and anchor to the destination point (the picnic table icon) AC > DF. By the Converse of the Hinge Theorem, is also congruent. We know that Connor's angle is larger than Rose's so, based on the Hinge Theorem, the distance that Connor has to travel to get to their ANSWER: destination point is further than Rose's. Therefore, m∠ABC > m∠DEF Rose is closer to the destination. ANSWER: 32. QT and RS Rose

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: QT > RS

ANSWER: false; orthocenter

ANSWER: true 4. To find the centroid of a triangle, first construct the angle bisectors. SOLUTION: To find the centroid of a triangle, first construct the medians. The sentence is false. The true sentence is "To find the centroid of a triangle, first construct the State whether each sentence is true or false . If medians." false, replace the underlined term to make a true sentence. ANSWER: 1. The altitudes of a triangle intersect at the centroid. false; medians SOLUTION: 5. The perpendicular bisectors of a triangle are The centroid is the the point where the medians concurrent lines. intersect. The orthocenter is the point where the altitudes intersect. false; orthocenter SOLUTION: The perpendicular bisectors of a triangle are ANSWER: concurrent lines. The statement is true. false; orthocenter ANSWER: 2. The point of concurrency of the medians of a triangle true is called the incenter. 6. A proof by contradiction uses indirect reasoning. SOLUTION: SOLUTION: The point where the medians intersect is the Indirect reasoning is key when writing a proof by centroid. The point of concurrency of the angle contradiction. The statement is true. bisectors of a triangleis called the incenter. The sentence is false. "The point of concurrency of the ANSWER: angle bisectors of a triangle is called the incenter." is true the true sentence. 7. A median of a triangle connects the midpoint of one ANSWER: side of the triangle to the midpoint of another side of false; angle bisectors the triangle. 3. The circumcenter of a triangle is equidistant from the SOLUTION: vertices of the triangle. A median of a triangle connects the vertex to the SOLUTION: midpoint of the side opposite it. The sentence is false. The true sentence is "A median of a triangle The point that is equidistant from the vertices of a connects the midpoint of one side of the triangle to triangle is called the circumcenter. The statement is the vertex opposite that side." true. ANSWER: ANSWER: false; the vertex opposite that side true 8. The incenter is the point at which the angle bisectors 4. To find the centroid of a triangle, first construct the of a triangle intersect. angle bisectors. SOLUTION: SOLUTION: The point where the angle bisectors intersect is To find the centroid of a triangle, first construct the called the incenter. The statement is true. medians. The sentence is false. The true sentence is "To find the centroid of a triangle, first construct the ANSWER: medians." true ANSWER: 9. Explain how to write a proof by contradiction. false; medians SOLUTION: 5. The perpendicular bisectors of a triangle are Assume that the conclusion is false and show that concurrent lines. this assumption leads to a statement that cannot be SOLUTION: true. The perpendicular bisectors of a triangle are ANSWER: concurrent lines. The statement is true. Assume that the conclusion is false and show that ANSWER: this assumption leads to a statement that cannot be true. true 10. Explain how to locate the largest angle in a scalene 6. A proof by contradiction uses indirect reasoning. triangle. Then explain when a triangle does not have SOLUTION: one largest angle. Indirect reasoning is key when writing a proof by SOLUTION: contradiction. The statement is true. The largest angle in a scalene triangle is opposite the ANSWER: longest side. In an isosceles triangle, there may be true two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are 7. A median of a triangle connects the midpoint of one longer than the base. In an equilateral triangle, all side of the triangle to the midpoint of another side of angles are the same size. the triangle. ANSWER: SOLUTION: The largest angle in a scalene triangle is opposite the A median of a triangle connects the vertex to the longest side. In an isosceles triangle, there may be midpoint of the side opposite it. The sentence is false. two congruent angles that are larger than the third The true sentence is "A median of a triangle angle, so the sides opposite the congruent angles are connects the midpoint of one side of the triangle to longer than the base. In an equilateral triangle, all the vertex opposite that side." angles are the same size.

ANSWER: 11. Find EG if G is the incenter of . false; the vertex opposite that side 8. The incenter is the point at which the angle bisectors of a triangle intersect. SOLUTION: The point where the angle bisectors intersect is SOLUTION: called the incenter. The statement is true. By the Incenter Theorem, since G is equidistant from the sides of , EG = FG. Find FG using the ANSWER: Pythagorean Theorem. true 9. Explain how to write a proof by contradiction. SOLUTION: Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true.

ANSWER: Since length cannot be negative, use only the positive Assume that the conclusion is false and show that square root, 5. this assumption leads to a statement that cannot be Since EG = FG, EG = 5. true. ANSWER: 10. Explain how to locate the largest angle in a scalene 5 triangle. Then explain when a triangle does not have one largest angle. Find each measure. 12. RS SOLUTION: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all angles are the same size. SOLUTION: Here RT = TS. By the converse of the Perpendicular ANSWER: The largest angle in a scalene triangle is opposite the Bisector Theorem, is a perpendicular bisector of longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third Therefore, . angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all angles are the same size.

11. Find EG if G is the incenter of . ANSWER: 9

13. XZ

SOLUTION: By the Incenter Theorem, since G is equidistant from the sides of , EG = FG. Find FG using the Pythagorean Theorem.

SOLUTION: From the figure, Thus,

Substitute y = 8 in XZ. Since length cannot be negative, use only the positive square root, 5. Since EG = FG, EG = 5.

ANSWER: 5 ANSWER: Find each measure. 34 12. RS 14. BASEBALL Jackson, Trevor, and Scott are warming up before a baseball game. One of their warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players? SOLUTION: Here RT = TS. By the converse of the Perpendicular Bisector Theorem, is a perpendicular bisector of

Therefore, .

SOLUTION: The players can be represented by the vertices of a triangle. The point that is equidistant from each ANSWER: vertex is called the circumcenter. Find the 9 circumcenter by constructing the perpendicular bisector of each side of the triangle. 13. XZ

SOLUTION:

From the figure, ANSWER: Thus,

Substitute y = 8 in XZ.

ANSWER: 34 15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . 14. BASEBALL Jackson, Trevor, and Scott are SOLUTION: warming up before a baseball game. One of their warm-up drills requires three players to form a The slope of is or So, the slope of triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players? the altitude, which is perpendicular to is . Now, the equation of the altitude from D to is:

In the same way, we can find the equation of the SOLUTION:

The players can be represented by the vertices of a altitude from E to triangle. The point that is equidistant from each The slope of is . So, the slope of the vertex is called the circumcenter. Find the altitude, which is perpendicular to is –2. circumcenter by constructing the perpendicular bisector of each side of the triangle. The equation of the altitude is

Solve the equations to find the intersection point of the altitudes.

ANSWER:

Substitute the value of x in one of the equations to find the y-coordinate.

So, the coordinates of the orthocenter of is . 15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

The slope of is or So, the slope of

the altitude, which is perpendicular to is .

Now, the equation of the altitude from D to is:

ANSWER: (2, 3)

16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the In the same way, we can find the equation of the ceiling so that they are parallel to the floor. She altitude from E to sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle The slope of is . So, the slope of the is to be hung by one chain, what are the coordinates altitude, which is perpendicular to is –2. of the point where the chain should attach to the triangle? The equation of the altitude is SOLUTION: In order for the triangles to hang so that they are balanced parallel to the floor, each triangle must be Solve the equations to find the intersection point of attached to its chain at its centroid. This point is the altitudes. located at the intersection of the medians of the triangle.

The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4). Substitute the value of x in one of the equations to find the y-coordinate. One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection So, the coordinates of the orthocenter of is of x = 3 and y = 4, and the location of the traingle’s

. centroid, is the point (3, 4).

ChapterANSWER: 5 Study Guide and Review ANSWER: (2, 3) (3, 4)

16. PROM Georgia is on the prom committee. She List the angles and sides of each triangle in wants to hang a dozen congruent triangles from the order from smallest to largest. ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates of the point where the chain should attach to the triangle? SOLUTION: 17. In order for the triangles to hang so that they are SOLUTION: balanced parallel to the floor, each triangle must be The sides from shortest to longest are . attached to its chain at its centroid. This point is located at the intersection of the medians of the The angles opposite these sides are ∠S, ∠R, and triangle. ∠T, respectively. So the angles from smallest to largest are ∠S, ∠R, and ∠T. The midpoint of the side from (0, 4) to (6, 0) is ANSWER: or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the 18. line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s SOLUTION: centroid, is the point (3, 4). Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle.

ANSWER: Replace x with 5.6 to find angle measures. (3, 4) List the angles and sides of each triangle in order from smallest to largest.

17. eSolutions Manual - Powered by Cognero Page 4 SOLUTION: The sides from shortest to longest are . The angles from smallest to largest are ∠N, ∠L, The angles opposite these sides are ∠S, ∠R, and ∠M. The sides opposite these angles are ∠T, respectively. So the angles from smallest to , respectively. So, the sides largest are ∠S, ∠R, and ∠T. from shortest to longest are . ANSWER: ANSWER: ∠N, ∠L, ∠M; , ,

19. NEIGHBORHOODS Anna, Sarah, and Irene live at the intersections of the three roads that make the triangle shown. If the girls want to spend the afternoon together, is it a shorter path for Anna to stop and get Sarah and go on to Irene’s house, or for Sarah to stop and get Irene and then go on to Anna’s

18. house? SOLUTION: Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle.

SOLUTION: The girls' houses can be represented by the vertices of a triangle. List the sides of the triangle in order Replace x with 5.6 to find angle measures. from shortest to longest. First find the missing angle measure using the Triangle Angle-Sum Theorem.

m∠Irene = 180 – (37 + 53) or 90

ANSWER:

The angles from smallest to largest are ∠N, ∠L, The shorter path is for Sarah to get Irene and then go ∠M. The sides opposite these angles are to Anna’s house. , respectively. So, the sides from shortest to longest are . State the assumption you would make to start an indirect proof of each statement. ANSWER: 20. ∠N, ∠L, ∠M; , , SOLUTION: To start an indirect proof, first assume that what you NEIGHBORHOODS 19. Anna, Sarah, and Irene live are trying to prove is false. at the intersections of the three roads that make the triangle shown. If the girls want to spend the ANSWER: afternoon together, is it a shorter path for Anna to

stop and get Sarah and go on to Irene’s house, or for m∠A < m∠B Sarah to stop and get Irene and then go on to Anna’s house? 21. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. is not congruent to .

ANSWER: is not congruent to .

22. is a right triangle. SOLUTION: SOLUTION: The girls' houses can be represented by the vertices To start an indirect proof, first assume that what you of a triangle. List the sides of the triangle in order are trying to prove is false. is not a right from shortest to longest. First find the missing angle triangle. measure using the Triangle Angle-Sum Theorem. ANSWER: m∠Irene = 180 – (37 + 53) or 90 is not a right triangle.

So, the angles from smallest to largest are ∠Anna, 23. If 3y < 12, then y < 4. ∠Sarah, ∠Irene. The sides opposite these angles are SOLUTION: the path from Sarah to Irene, the path from Irene to Anna, and the path from Sarah to Anna, respectively. To start an indirect proof, first assume that what you So, the shorter path is for Sarah to get Irene and then are trying to prove is false. If 3y < 12, then y ≥ 4. go to Anna’s house. ANSWER: ANSWER: The shorter path is for Sarah to get Irene and then go 24. Write an indirect proof to show that if two angles are to Anna’s house. complementary, neither angle is a right angle. State the assumption you would make to start SOLUTION: an indirect proof of each statement. To start an indirect proof, first assume that what you 20. are trying to prove is false. In this case, try to find a contradiction if you assume that x or y are right SOLUTION: angles. To start an indirect proof, first assume that what you are trying to prove is false. Let the measure of one angle be x and the measure of the other angle be y. By the definition of ANSWER: complementary angles, x + y = 90. m∠A < m∠B Step 1 Assume that the angle with the measure x is a right angle. Then x = 90. 21. SOLUTION: Step 2 Since x = 90, then x + y > 90. This is a contradiction because we know that x + y = 90. To start an indirect proof, first assume that what you

are trying to prove is false. is not congruent Step 3 Since the assumption that one angle is a right to . angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle ANSWER: is a right angle must be true. is not congruent to . ANSWER: 22. is a right triangle. Let the measure of one angle be x and the measure SOLUTION: of the other angle be y. By the definition of To start an indirect proof, first assume that what you complementary angles, x + y = 90. are trying to prove is false. is not a right Step 1 Assume that the angle with the measure x is a triangle. right angle. Then x = 90. ANSWER: Step 2 Since x = 90, then x + y > 90. This is a is not a right triangle. contradiction because we know that x + y = 90. 23. If 3y < 12, then y < 4. Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must SOLUTION: To start an indirect proof, first assume that what you be false. Therefore, the conclusion that neither angle are trying to prove is false. If 3y < 12, then y ≥ 4. is a right angle must be true.

ANSWER: 25. CONCESSIONS Isaac purchased two items at the concession stand at the Houston Dynamo game and spent over $10. Use indirect reasoning to show that 24. Write an indirect proof to show that if two angles are at least one of the items he purchased was over $5. complementary, neither angle is a right angle. SOLUTION: SOLUTION: To start an indirect proof, first assume that what you To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a are trying to prove is false. In this case, try to find a contradiction if you assume that the cost of item x contradiction if you assume that x or y are right and the cost of item y are less than or equal to $5. angles. Let the cost of one item be x, and the cost of the Let the measure of one angle be x and the measure other item be y. of the other angle be y. By the definition of Given: x + y > 10 complementary angles, x + y = 90. Prove: x > 5 or y > 5 Indirect Proof: Step 1 Assume that the angle with the measure x is a Step 1 Assume that and . right angle. Then x = 90.

Step 2 If and , then or Step 2 Since x = 90, then x + y > 90. This is a . This is a contradiction because we contradiction because we know that x + y = 90. know that x + y > 50.

Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must Step 3 Since the assumption that and be false. Therefore, the conclusion that neither angle leads to a contradiction of a known fact, the is a right angle must be true. assumption must be false. Therefore, the conclusion that x > 5 or y > 5 must be true. Thus, at least one ANSWER: item had to be over $5. Let the measure of one angle be x and the measure ANSWER: of the other angle be y. By the definition of Let the cost of one item be x, and the cost of the complementary angles, x + y = 90. other item be y. Step 1 Assume that the angle with the measure x is a Given: x + y > 10

right angle. Then x = 90. Prove: x > 5 or y > 5 Step 2 Since x = 90, then x + y > 90. This is a Indirect Proof:

contradiction because we know that x + y = 90. Step 1 Assume that and . Step 3 Since the assumption that one angle is a right Step 2 If and , then , or angle leads to a contradiction, the assumption must . This is a contradiction because we be false. Therefore, the conclusion that neither angle know that x + y > 10.

is a right angle must be true. Step 3 Since the assumption that x ≤ 5 and 25. CONCESSIONS Isaac purchased two items at the leads to a contradiction of a known fact, the concession stand at the Houston Dynamo game and assumption must be false. Therefore, the conclusion spent over $10. Use indirect reasoning to show that that x > 5 or y > 5 must be true. Thus, at least one at least one of the items he purchased was over $5. item had to be over $5. SOLUTION: To start an indirect proof, first assume that what you Is it possible to form a triangle with the given are trying to prove is false. In this case, try to find a lengths? If not, explain why not. contradiction if you assume that the cost of item x 26. 5, 6, 9 and the cost of item y are less than or equal to $5. SOLUTION:

Let the cost of one item be x, and the cost of the Check each inequality.

other item be y. 5 + 6 > 9 5 + 9 > 6 Given: x + y > 10 6 + 9 > 5 Prove: x > 5 or y > 5 Since the sum of each pair of side lengths is greater Indirect Proof: than the third side length, lengths of 5, 6, and 9 units Step 1 Assume that and . will form a triangle.

Step 2 If and , then or ANSWER: . This is a contradiction because we Yes know that x + y > 50.

Step 3 Since the assumption that and 27. 3, 4, 8 leads to a contradiction of a known fact, the SOLUTION: assumption must be false. Therefore, the conclusion 3 + 4 < 8 that x > 5 or y > 5 must be true. Thus, at least one Since the sum of one pair of side lengths is not

item had to be over $5. greater than the third side length, lengths 3, 4, and 8 ANSWER: units will not form a triangle. Let the cost of one item be x, and the cost of the ANSWER: other item be y. No; 3 + 4 < 8 Given: x + y > 10 Find the range for the measure of the third side Prove: x > 5 or y > 5 of a triangle given the measure of two sides. Indirect Proof: 28. 5 ft, 7 ft Step 1 Assume that and . SOLUTION: Step 2 If and , then , or Let n represent the length of the third side. . This is a contradiction because we According to the Triangle Inequality Theorem, the know that x + y > 10. largest side cannot be greater than the sum of the Step 3 Since the assumption that x ≤ 5 and other two sides. leads to a contradiction of a known fact, the If n is the largest side, then n must be less than 5 + assumption must be false. Therefore, the conclusion 7. Therefore, n < 12. that x > 5 or y > 5 must be true. Thus, at least one If n is not the largest side, then 7 is the largest and 7 item had to be over $5. must be less than 5 + n. Therefore, 2 < n.

Is it possible to form a triangle with the given Combining these two inequalities, we get 2 < n < 12. lengths? If not, explain why not. 26. 5, 6, 9 ANSWER: Let x be the length of the third side. 2 ft < x < 12 ft SOLUTION: Check each inequality. 5 + 6 > 9 29. 10.5 cm, 4 cm

5 + 9 > 6 SOLUTION: 6 + 9 > 5 Since the sum of each pair of side lengths is greater Let n represent the length of the third side. than the third side length, lengths of 5, 6, and 9 units will form a triangle. According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the ANSWER: other two sides. Yes If n is the largest side, then n must be less than 10.5 + 4. Therefore, n < 14.5. 27. 3, 4, 8 If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n. SOLUTION:

3 + 4 < 8 Combining these two inequalities, we get 6.5 < n < Since the sum of one pair of side lengths is not 14.5. greater than the third side length, lengths 3, 4, and 8 units will not form a triangle. ANSWER: ANSWER: Let x be the length of the third side. 6.5 cm < x < No; 3 + 4 < 8 14.5 cm.

Find the range for the measure of the third side 30. BIKES Leonard rides his bike to visit Josh. Since of a triangle given the measure of two sides. High Street is closed, he has to travel 2 miles down 28. 5 ft, 7 ft Main Street and turn to travel 3 miles farther on 5th Street. If the three streets form a triangle with SOLUTION: Leonard and Josh’s house as two of the vertices, Let n represent the length of the third side. find the range of the possible distance between Leonard and Josh’s houses when traveling straight According to the Triangle Inequality Theorem, the down High Street. largest side cannot be greater than the sum of the other two sides. SOLUTION: Let x be the distance between Leonard and Josh’s If n is the largest side, then n must be less than 5 + houses when traveling straight down High Street. 7. Therefore, n < 12. If n is not the largest side, then 7 is the largest and 7 Next, set up and solve each of the three triangle must be less than 5 + n. Therefore, 2 < n. inequalities. 2 + 3 > x, 2 + x > 3, and 3 + x > 2

Combining these two inequalities, we get 2 < n < 12. That is, 5 > x, x > 1, and x > –1.

ANSWER: Notice that x > –1 is always true for any whole number measure for x. Combining the two remaining Let x be the length of the third side. 2 ft < x < 12 ft inequalities, the range of values that fit both inequalities is x > 1 and x < 5, which can be written 29. 10.5 cm, 4 cm as 1 mile < x < 5 miles. Therefore, the distance is greater than 1 mile and less than 5 miles. SOLUTION: Let n represent the length of the third side. ANSWER: The distance is greater than 1 mile and less than 5 According to the Triangle Inequality Theorem, the miles. largest side cannot be greater than the sum of the other two sides. Compare the given measures.

31. m∠ABC, m∠DEF If n is the largest side, then n must be less than 10.5 + 4. Therefore, n < 14.5. If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n.

Combining these two inequalities, we get 6.5 < n < 14.5.

ANSWER: SOLUTION: Let x be the length of the third side. 6.5 cm < x < In and , and 14.5 cm. AC > DF. By the Converse of the Hinge Theorem,

30. BIKES Leonard rides his bike to visit Josh. Since High Street is closed, he has to travel 2 miles down ANSWER: Main Street and turn to travel 3 miles farther on 5th Street. If the three streets form a triangle with m∠ABC > m∠DEF Leonard and Josh’s house as two of the vertices, find the range of the possible distance between 32. QT and RS Leonard and Josh’s houses when traveling straight down High Street. SOLUTION: Let x be the distance between Leonard and Josh’s houses when traveling straight down High Street.

Next, set up and solve each of the three triangle inequalities. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1.

Notice that x > –1 is always true for any whole number measure for x. Combining the two remaining SOLUTION: inequalities, the range of values that fit both In and , inequalities is x > 1 and x < 5, which can be written and . as 1 mile < x < 5 miles. Therefore, the distance is By the Hinge Theorem, . greater than 1 mile and less than 5 miles. ANSWER: ANSWER: QT > RS The distance is greater than 1 mile and less than 5 miles. 33. BOATING Rose and Connor each row across a pond heading to the same point. Neither of them has Compare the given measures. rowed a boat before, so they both go off course as 31. m∠ABC, m∠DEF shown in the diagram. After two minutes, they have each traveled 50 yards. Who is closer to their destination?

SOLUTION: In and , and AC > DF. By the Converse of the Hinge Theorem,

ANSWER: SOLUTION:

m∠ABC > m∠DEF As indicated, the distance from the anchor icon to each boat is congruent and the distanced from the 32. QT and RS anchor to the destination point (the picnic table icon) is also congruent. We know that Connor's angle is larger than Rose's so, based on the Hinge Theorem, the distance that Connor has to travel to get to their destination point is further than Rose's. Therefore, Rose is closer to the destination.

ANSWER: Rose

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: QT > RS

ANSWER: false; orthocenter

ANSWER: true 4. To find the centroid of a triangle, first construct the angle bisectors. SOLUTION: To find the centroid of a triangle, first construct the medians. The sentence is false. The true sentence is State whether each sentence is true or false . If "To find the centroid of a triangle, first construct the false, replace the underlined term to make a medians." true sentence. 1. The altitudes of a triangle intersect at the centroid. ANSWER: SOLUTION: false; medians The centroid is the the point where the medians 5. The perpendicular bisectors of a triangle are intersect. The orthocenter is the point where the concurrent lines. altitudes intersect. false; orthocenter SOLUTION: ANSWER: The perpendicular bisectors of a triangle are false; orthocenter concurrent lines. The statement is true.

ANSWER: 2. The point of concurrency of the medians of a triangle is called the incenter. true SOLUTION: 6. A proof by contradiction uses indirect reasoning. The point where the medians intersect is the SOLUTION: centroid. The point of concurrency of the angle Indirect reasoning is key when writing a proof by bisectors of a triangleis called the incenter. The contradiction. The statement is true. sentence is false. "The point of concurrency of the angle bisectors of a triangle is called the incenter." is ANSWER: the true sentence. true ANSWER: 7. A median of a triangle connects the midpoint of one false; angle bisectors side of the triangle to the midpoint of another side of the triangle. 3. The circumcenter of a triangle is equidistant from the vertices of the triangle. SOLUTION: SOLUTION: A median of a triangle connects the vertex to the midpoint of the side opposite it. The sentence is false. The point that is equidistant from the vertices of a The true sentence is "A median of a triangle triangle is called the circumcenter. The statement is connects the midpoint of one side of the triangle to true. the vertex opposite that side." ANSWER: ANSWER: true false; the vertex opposite that side 4. To find the centroid of a triangle, first construct the 8. The incenter is the point at which the angle bisectors angle bisectors. of a triangle intersect. SOLUTION: SOLUTION: To find the centroid of a triangle, first construct the The point where the angle bisectors intersect is medians. The sentence is false. The true sentence is called the incenter. The statement is true. "To find the centroid of a triangle, first construct the medians." ANSWER: ANSWER: true false; medians 9. Explain how to write a proof by contradiction. 5. The perpendicular bisectors of a triangle are SOLUTION: concurrent lines. Assume that the conclusion is false and show that SOLUTION: this assumption leads to a statement that cannot be true. The perpendicular bisectors of a triangle are concurrent lines. The statement is true. ANSWER: ANSWER: Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true true. 6. A proof by contradiction uses indirect reasoning. 10. Explain how to locate the largest angle in a scalene SOLUTION: triangle. Then explain when a triangle does not have Indirect reasoning is key when writing a proof by one largest angle. contradiction. The statement is true. SOLUTION: ANSWER: The largest angle in a scalene triangle is opposite the true longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third 7. A median of a triangle connects the midpoint of one angle, so the sides opposite the congruent angles are side of the triangle to the midpoint of another side of longer than the base. In an equilateral triangle, all the triangle. angles are the same size. SOLUTION: ANSWER: A median of a triangle connects the vertex to the The largest angle in a scalene triangle is opposite the midpoint of the side opposite it. The sentence is false. longest side. In an isosceles triangle, there may be The true sentence is "A median of a triangle two congruent angles that are larger than the third connects the midpoint of one side of the triangle to angle, so the sides opposite the congruent angles are the vertex opposite that side." longer than the base. In an equilateral triangle, all angles are the same size. ANSWER: false; the vertex opposite that side 11. Find EG if G is the incenter of . 8. The incenter is the point at which the angle bisectors of a triangle intersect. SOLUTION:

The point where the angle bisectors intersect is called the incenter. The statement is true. SOLUTION: By the Incenter Theorem, since G is equidistant from ANSWER: the sides of , EG = FG. Find FG using the true Pythagorean Theorem.

9. Explain how to write a proof by contradiction. SOLUTION: Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true.

ANSWER: Assume that the conclusion is false and show that Since length cannot be negative, use only the positive this assumption leads to a statement that cannot be square root, 5. true. Since EG = FG, EG = 5.

10. Explain how to locate the largest angle in a scalene ANSWER: triangle. Then explain when a triangle does not have 5 one largest angle. Find each measure. SOLUTION: 12. RS The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all angles are the same size. SOLUTION: ANSWER: Here RT = TS. By the converse of the Perpendicular The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be Bisector Theorem, is a perpendicular bisector of two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are Therefore, . longer than the base. In an equilateral triangle, all angles are the same size.

11. Find EG if G is the incenter of . ANSWER: 9

13. XZ SOLUTION: By the Incenter Theorem, since G is equidistant from the sides of , EG = FG. Find FG using the Pythagorean Theorem.

SOLUTION: From the figure, Thus,

Since length cannot be negative, use only the positive Substitute y = 8 in XZ. square root, 5. Since EG = FG, EG = 5.

ANSWER: 5 ANSWER: Find each measure. 12. RS 34 14. BASEBALL Jackson, Trevor, and Scott are warming up before a baseball game. One of their warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players? SOLUTION: Here RT = TS. By the converse of the Perpendicular Bisector Theorem, is a perpendicular bisector of

Therefore, .

SOLUTION: The players can be represented by the vertices of a ANSWER: triangle. The point that is equidistant from each 9 vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular bisector of each side of the triangle. 13. XZ

SOLUTION:

From the figure, Thus, ANSWER:

Substitute y = 8 in XZ.

ANSWER: 34 15. The vertices of are D(0, 0), E(0, 7), and F(6, 14. BASEBALL Jackson, Trevor, and Scott are 3). Find the coordinates of the orthocenter of . warming up before a baseball game. One of their SOLUTION: warm-up drills requires three players to form a triangle, with one player in the middle. Where should The slope of is or So, the slope of the fourth player stand so that he is the same distance from the other three players? the altitude, which is perpendicular to is .

Now, the equation of the altitude from D to is:

SOLUTION: In the same way, we can find the equation of the The players can be represented by the vertices of a triangle. The point that is equidistant from each altitude from E to vertex is called the circumcenter. Find the The slope of is . So, the slope of the circumcenter by constructing the perpendicular altitude, which is perpendicular to is –2. bisector of each side of the triangle.

The equation of the altitude is

Solve the equations to find the intersection point of the altitudes.

ANSWER:

Substitute the value of x in one of the equations to find the y-coordinate.

So, the coordinates of the orthocenter of is . 15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

The slope of is or So, the slope of

the altitude, which is perpendicular to is .

Now, the equation of the altitude from D to is:

ANSWER: (2, 3)

16. PROM Georgia is on the prom committee. She In the same way, we can find the equation of the wants to hang a dozen congruent triangles from the altitude from E to ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with The slope of is . So, the slope of the coordinates (0, 4), (3, 8), and (6, 0). If each triangle altitude, which is perpendicular to is –2. is to be hung by one chain, what are the coordinates of the point where the chain should attach to the

The equation of the altitude is triangle? SOLUTION: In order for the triangles to hang so that they are Solve the equations to find the intersection point of balanced parallel to the floor, each triangle must be the altitudes. attached to its chain at its centroid. This point is located at the intersection of the medians of the triangle.

The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

Substitute the value of x in one of the equations to from (3, 8) to (6, 0) is or (4.5, 4). find the y-coordinate.

. of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4).

ANSWER: (2, 3) ANSWER: (3, 4) 16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the List the angles and sides of each triangle in ceiling so that they are parallel to the floor. She order from smallest to largest. sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates of the point where the chain should attach to the triangle? SOLUTION:

In order for the triangles to hang so that they are 17. balanced parallel to the floor, each triangle must be SOLUTION: attached to its chain at its centroid. This point is The sides from shortest to longest are . located at the intersection of the medians of the triangle. The angles opposite these sides are ∠S, ∠R, and ∠T, respectively. So the angles from smallest to The midpoint of the side from (0, 4) to (6, 0) is largest are ∠S, ∠R, and ∠T. or (3, 2). The midpoint of the side ANSWER:

from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection 18. of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4). SOLUTION: Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle.

ANSWER: (3, 4) Replace x with 5.6 to find angle measures. List the angles and sides of each triangle in order from smallest to largest.

17. SOLUTION: The sides from shortest to longest are . The angles opposite these sides are ∠S, ∠R, and The angles from smallest to largest are ∠N, ∠L, ∠T, respectively. So the angles from smallest to ∠M. The sides opposite these angles are , respectively. So, the sides largest are ∠S, ∠R, and ∠T. from shortest to longest are . ANSWER: Chapter 5 Study Guide and Review ANSWER: ∠N, ∠L, ∠M; , ,

19. NEIGHBORHOODS Anna, Sarah, and Irene live at the intersections of the three roads that make the triangle shown. If the girls want to spend the afternoon together, is it a shorter path for Anna to stop and get Sarah and go on to Irene’s house, or for 18. Sarah to stop and get Irene and then go on to Anna’s house? SOLUTION: Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle.

SOLUTION: The girls' houses can be represented by the vertices Replace x with 5.6 to find angle measures. of a triangle. List the sides of the triangle in order from shortest to longest. First find the missing angle measure using the Triangle Angle-Sum Theorem.

m∠Irene = 180 – (37 + 53) or 90

ANSWER: The angles from smallest to largest are ∠N, ∠L, The shorter path is for Sarah to get Irene and then go ∠M. The sides opposite these angles are , respectively. So, the sides to Anna’s house. from shortest to longest are . State the assumption you would make to start ANSWER: an indirect proof of each statement. 20. ∠N, ∠L, ∠M; , , SOLUTION: 19. NEIGHBORHOODS Anna, Sarah, and Irene live To start an indirect proof, first assume that what you at the intersections of the three roads that make the are trying to prove is false. triangle shown. If the girls want to spend the afternoon together, is it a shorter path for Anna to ANSWER: stop and get Sarah and go on to Irene’s house, or for

Sarah to stop and get Irene and then go on to Anna’s m∠A < m∠B house? 21. SOLUTION: eSolutions Manual - Powered by Cognero To start an indirect proof, first assume that whatPage you 5 are trying to prove is false. is not congruent to .

ANSWER: is not congruent to .

SOLUTION: 22. is a right triangle. The girls' houses can be represented by the vertices SOLUTION: of a triangle. List the sides of the triangle in order To start an indirect proof, first assume that what you from shortest to longest. First find the missing angle are trying to prove is false. is not a right

measure using the Triangle Angle-Sum Theorem. triangle.

m∠Irene = 180 – (37 + 53) or 90 ANSWER: is not a right triangle. So, the angles from smallest to largest are ∠Anna, ∠Sarah, ∠Irene. The sides opposite these angles are 23. If 3y < 12, then y < 4. the path from Sarah to Irene, the path from Irene to SOLUTION: Anna, and the path from Sarah to Anna, respectively. So, the shorter path is for Sarah to get Irene and then To start an indirect proof, first assume that what you go to Anna’s house. are trying to prove is false. If 3y < 12, then y ≥ 4.

ANSWER: ANSWER: The shorter path is for Sarah to get Irene and then go to Anna’s house. 24. Write an indirect proof to show that if two angles are complementary, neither angle is a right angle. State the assumption you would make to start an indirect proof of each statement. SOLUTION: 20. To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a SOLUTION: contradiction if you assume that x or y are right To start an indirect proof, first assume that what you angles. are trying to prove is false. Let the measure of one angle be x and the measure ANSWER: of the other angle be y. By the definition of m∠A < m∠B complementary angles, x + y = 90.

Step 1 Assume that the angle with the measure x is a 21. right angle. Then x = 90. SOLUTION: Step 2 Since x = 90, then x + y > 90. This is a To start an indirect proof, first assume that what you contradiction because we know that x + y = 90. are trying to prove is false. is not congruent to . Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must ANSWER: be false. Therefore, the conclusion that neither angle is not congruent to . is a right angle must be true.

22. is a right triangle. ANSWER: SOLUTION: Let the measure of one angle be x and the measure To start an indirect proof, first assume that what you of the other angle be y. By the definition of are trying to prove is false. is not a right complementary angles, x + y = 90. triangle. Step 1 Assume that the angle with the measure x is a ANSWER: right angle. Then x = 90. is not a right triangle. Step 2 Since x = 90, then x + y > 90. This is a contradiction because we know that x + y = 90. 23. If 3y < 12, then y < 4. Step 3 Since the assumption that one angle is a right SOLUTION: angle leads to a contradiction, the assumption must To start an indirect proof, first assume that what you are trying to prove is false. If 3y < 12, then y ≥ 4. be false. Therefore, the conclusion that neither angle is a right angle must be true. ANSWER: 25. CONCESSIONS Isaac purchased two items at the concession stand at the Houston Dynamo game and 24. Write an indirect proof to show that if two angles are spent over $10. Use indirect reasoning to show that complementary, neither angle is a right angle. at least one of the items he purchased was over $5. SOLUTION: SOLUTION: To start an indirect proof, first assume that what you To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a are trying to prove is false. In this case, try to find a contradiction if you assume that x or y are right contradiction if you assume that the cost of item x angles. and the cost of item y are less than or equal to $5.

Let the measure of one angle be x and the measure Let the cost of one item be x, and the cost of the of the other angle be y. By the definition of other item be y. complementary angles, x + y = 90. Given: x + y > 10 Prove: x > 5 or y > 5 Step 1 Assume that the angle with the measure x is a Indirect Proof: right angle. Then x = 90. Step 1 Assume that and .

Step 2 Since x = 90, then x + y > 90. This is a Step 2 If and , then or contradiction because we know that x + y = 90. . This is a contradiction because we

know that x + y > 50. Step 3 Since the assumption that one angle is a right

angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle Step 3 Since the assumption that and is a right angle must be true. leads to a contradiction of a known fact, the assumption must be false. Therefore, the conclusion ANSWER: that x > 5 or y > 5 must be true. Thus, at least one

Let the measure of one angle be x and the measure item had to be over $5. of the other angle be y. By the definition of ANSWER: complementary angles, x + y = 90. Let the cost of one item be x, and the cost of the Step 1 Assume that the angle with the measure x is a other item be y. right angle. Then x = 90. Given: x + y > 10 Step 2 Since x = 90, then x + y > 90. This is a Prove: x > 5 or y > 5 contradiction because we know that x + y = 90. Indirect Proof: Step 3 Since the assumption that one angle is a right Step 1 Assume that and . angle leads to a contradiction, the assumption must Step 2 If and , then , or be false. Therefore, the conclusion that neither angle . This is a contradiction because we is a right angle must be true. know that x + y > 10. Step 3 Since the assumption that x ≤ 5 and 25. CONCESSIONS Isaac purchased two items at the concession stand at the Houston Dynamo game and leads to a contradiction of a known fact, the spent over $10. Use indirect reasoning to show that assumption must be false. Therefore, the conclusion at least one of the items he purchased was over $5. that x > 5 or y > 5 must be true. Thus, at least one SOLUTION: item had to be over $5. To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a Is it possible to form a triangle with the given contradiction if you assume that the cost of item x lengths? If not, explain why not. and the cost of item y are less than or equal to $5. 26. 5, 6, 9

SOLUTION: Let the cost of one item be x, and the cost of the other item be y. Check each inequality. 5 + 6 > 9 Given: x + y > 10 5 + 9 > 6 Prove: x > 5 or y > 5 6 + 9 > 5 Indirect Proof: Since the sum of each pair of side lengths is greater Step 1 Assume that and . than the third side length, lengths of 5, 6, and 9 units will form a triangle. Step 2 If and , then or . This is a contradiction because we ANSWER: know that x + y > 50. Yes

Step 3 Since the assumption that and leads to a contradiction of a known fact, the 27. 3, 4, 8 assumption must be false. Therefore, the conclusion SOLUTION: that x > 5 or y > 5 must be true. Thus, at least one 3 + 4 < 8

item had to be over $5. Since the sum of one pair of side lengths is not ANSWER: greater than the third side length, lengths 3, 4, and 8 units will not form a triangle. Let the cost of one item be x, and the cost of the other item be y. ANSWER: Given: x + y > 10 No; 3 + 4 < 8 Prove: x > 5 or y > 5 Find the range for the measure of the third side Indirect Proof: of a triangle given the measure of two sides. Step 1 Assume that and . 28. 5 ft, 7 ft Step 2 If and , then , or SOLUTION: . This is a contradiction because we Let n represent the length of the third side.

know that x + y > 10. According to the Triangle Inequality Theorem, the Step 3 Since the assumption that x ≤ 5 and largest side cannot be greater than the sum of the leads to a contradiction of a known fact, the other two sides.

assumption must be false. Therefore, the conclusion If n is the largest side, then n must be less than 5 + that x > 5 or y > 5 must be true. Thus, at least one 7. Therefore, n < 12. item had to be over $5. If n is not the largest side, then 7 is the largest and 7 must be less than 5 + n. Therefore, 2 < n. Is it possible to form a triangle with the given lengths? If not, explain why not. Combining these two inequalities, we get 2 < n < 12. 26. 5, 6, 9 ANSWER: SOLUTION: Let x be the length of the third side. 2 ft < x < 12 ft Check each inequality. 5 + 6 > 9 5 + 9 > 6 29. 10.5 cm, 4 cm

6 + 9 > 5 SOLUTION: Since the sum of each pair of side lengths is greater than the third side length, lengths of 5, 6, and 9 units Let n represent the length of the third side. will form a triangle. According to the Triangle Inequality Theorem, the ANSWER: largest side cannot be greater than the sum of the

Yes other two sides.

If n is the largest side, then n must be less than 10.5 27. 3, 4, 8 + 4. Therefore, n < 14.5. If n is not the largest side, then 10.5 is the largest and SOLUTION: 10.5 must be less than 4 + n. Therefore, 6.5 < n. 3 + 4 < 8 Since the sum of one pair of side lengths is not Combining these two inequalities, we get 6.5 < n < greater than the third side length, lengths 3, 4, and 8 14.5. units will not form a triangle. ANSWER: ANSWER: Let x be the length of the third side. 6.5 cm < x < No; 3 + 4 < 8 14.5 cm. Find the range for the measure of the third side of a triangle given the measure of two sides. 30. BIKES Leonard rides his bike to visit Josh. Since 28. 5 ft, 7 ft High Street is closed, he has to travel 2 miles down Main Street and turn to travel 3 miles farther on 5th SOLUTION: Street. If the three streets form a triangle with Let n represent the length of the third side. Leonard and Josh’s house as two of the vertices, find the range of the possible distance between According to the Triangle Inequality Theorem, the Leonard and Josh’s houses when traveling straight largest side cannot be greater than the sum of the down High Street. other two sides. SOLUTION: If n is the largest side, then n must be less than 5 + Let x be the distance between Leonard and Josh’s 7. Therefore, n < 12. houses when traveling straight down High Street. If n is not the largest side, then 7 is the largest and 7 must be less than 5 + n. Therefore, 2 < n. Next, set up and solve each of the three triangle inequalities.

Combining these two inequalities, we get 2 < n < 12. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1. ANSWER: Notice that x > –1 is always true for any whole Let x be the length of the third side. 2 ft < x < 12 ft number measure for x. Combining the two remaining inequalities, the range of values that fit both 29. 10.5 cm, 4 cm inequalities is x > 1 and x < 5, which can be written as 1 mile < x < 5 miles. Therefore, the distance is SOLUTION: greater than 1 mile and less than 5 miles. Let n represent the length of the third side. ANSWER: According to the Triangle Inequality Theorem, the The distance is greater than 1 mile and less than 5 largest side cannot be greater than the sum of the miles. other two sides. Compare the given measures. If n is the largest side, then n must be less than 10.5 31. m∠ABC, m∠DEF + 4. Therefore, n < 14.5. If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n.

Combining these two inequalities, we get 6.5 < n < 14.5.

ANSWER: Let x be the length of the third side. 6.5 cm < x < SOLUTION: 14.5 cm. In and , and AC > DF. By the Converse of the Hinge Theorem, BIKES 30. Leonard rides his bike to visit Josh. Since High Street is closed, he has to travel 2 miles down Main Street and turn to travel 3 miles farther on 5th ANSWER: Street. If the three streets form a triangle with Leonard and Josh’s house as two of the vertices, m∠ABC > m∠DEF find the range of the possible distance between Leonard and Josh’s houses when traveling straight 32. QT and RS down High Street. SOLUTION: Let x be the distance between Leonard and Josh’s houses when traveling straight down High Street.

Next, set up and solve each of the three triangle inequalities. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1.

Notice that x > –1 is always true for any whole number measure for x. Combining the two remaining inequalities, the range of values that fit both SOLUTION: inequalities is x > 1 and x < 5, which can be written In and , as 1 mile < x < 5 miles. Therefore, the distance is and . greater than 1 mile and less than 5 miles. By the Hinge Theorem, .

ANSWER: ANSWER: The distance is greater than 1 mile and less than 5 QT > RS miles. 33. BOATING Rose and Connor each row across a Compare the given measures. pond heading to the same point. Neither of them has 31. m∠ABC, m∠DEF rowed a boat before, so they both go off course as shown in the diagram. After two minutes, they have each traveled 50 yards. Who is closer to their destination?

SOLUTION: In and , and AC > DF. By the Converse of the Hinge Theorem,

ANSWER: m∠ABC > m∠DEF SOLUTION: As indicated, the distance from the anchor icon to 32. QT and RS each boat is congruent and the distanced from the anchor to the destination point (the picnic table icon) is also congruent. We know that Connor's angle is larger than Rose's so, based on the Hinge Theorem, the distance that Connor has to travel to get to their destination point is further than Rose's. Therefore, Rose is closer to the destination.

ANSWER: Rose

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: QT > RS

ANSWER: false; orthocenter

ANSWER: false; angle bisectors 3. The circumcenter of a triangle is equidistant from the vertices of the triangle. State whether each sentence is true or false . If SOLUTION: false, replace the underlined term to make a The point that is equidistant from the vertices of a true sentence. triangle is called the circumcenter. The statement is 1. The altitudes of a triangle intersect at the centroid. true. SOLUTION: ANSWER: The centroid is the the point where the medians true intersect. The orthocenter is the point where the altitudes intersect. false; orthocenter 4. To find the centroid of a triangle, first construct the angle bisectors. ANSWER: false; orthocenter SOLUTION: To find the centroid of a triangle, first construct the 2. The point of concurrency of the medians of a triangle medians. The sentence is false. The true sentence is is called the incenter. "To find the centroid of a triangle, first construct the medians." SOLUTION: The point where the medians intersect is the ANSWER: centroid. The point of concurrency of the angle false; medians bisectors of a triangleis called the incenter. The sentence is false. "The point of concurrency of the 5. The perpendicular bisectors of a triangle are angle bisectors of a triangle is called the incenter." is concurrent lines. the true sentence. SOLUTION: ANSWER: The perpendicular bisectors of a triangle are concurrent lines. The statement is true. false; angle bisectors ANSWER: 3. The circumcenter of a triangle is equidistant from the vertices of the triangle. true SOLUTION: 6. A proof by contradiction uses indirect reasoning. The point that is equidistant from the vertices of a SOLUTION: triangle is called the circumcenter. The statement is Indirect reasoning is key when writing a proof by true. contradiction. The statement is true. ANSWER: ANSWER: true true 4. To find the centroid of a triangle, first construct the 7. A median of a triangle connects the midpoint of one angle bisectors. side of the triangle to the midpoint of another side of SOLUTION: the triangle. To find the centroid of a triangle, first construct the SOLUTION: medians. The sentence is false. The true sentence is A median of a triangle connects the vertex to the "To find the centroid of a triangle, first construct the midpoint of the side opposite it. The sentence is false. medians." The true sentence is "A median of a triangle ANSWER: connects the midpoint of one side of the triangle to the vertex opposite that side." false; medians ANSWER: 5. The perpendicular bisectors of a triangle are concurrent lines. false; the vertex opposite that side SOLUTION: 8. The incenter is the point at which the angle bisectors The perpendicular bisectors of a triangle are of a triangle intersect. concurrent lines. The statement is true. SOLUTION: ANSWER: The point where the angle bisectors intersect is called the incenter. The statement is true. true ANSWER: 6. A proof by contradiction uses indirect reasoning. true SOLUTION: Indirect reasoning is key when writing a proof by 9. Explain how to write a proof by contradiction. contradiction. The statement is true. SOLUTION: ANSWER: Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true true. 7. A median of a triangle connects the midpoint of one ANSWER: side of the triangle to the midpoint of another side of the triangle. Assume that the conclusion is false and show that this assumption leads to a statement that cannot be SOLUTION: true. A median of a triangle connects the vertex to the midpoint of the side opposite it. The sentence is false. 10. Explain how to locate the largest angle in a scalene The true sentence is "A median of a triangle triangle. Then explain when a triangle does not have connects the midpoint of one side of the triangle to one largest angle. the vertex opposite that side." SOLUTION: ANSWER: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be false; the vertex opposite that side two congruent angles that are larger than the third 8. The incenter is the point at which the angle bisectors angle, so the sides opposite the congruent angles are of a triangle intersect. longer than the base. In an equilateral triangle, all angles are the same size. SOLUTION: The point where the angle bisectors intersect is ANSWER: called the incenter. The statement is true. The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be ANSWER: two congruent angles that are larger than the third true angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all 9. Explain how to write a proof by contradiction. angles are the same size. SOLUTION: 11. Find EG if G is the incenter of . Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true.

ANSWER:

Assume that the conclusion is false and show that this assumption leads to a statement that cannot be SOLUTION: true. By the Incenter Theorem, since G is equidistant from the sides of , EG = FG. Find FG using the 10. Explain how to locate the largest angle in a scalene Pythagorean Theorem. triangle. Then explain when a triangle does not have one largest angle. SOLUTION: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all angles are the same size. Since length cannot be negative, use only the positive square root, 5. ANSWER: Since EG = FG, EG = 5. The largest angle in a scalene triangle is opposite the ANSWER: longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third 5 angle, so the sides opposite the congruent angles are Find each measure. longer than the base. In an equilateral triangle, all 12. RS angles are the same size.

11. Find EG if G is the incenter of .

SOLUTION: Here RT = TS. By the converse of the Perpendicular SOLUTION: Bisector Theorem, is a perpendicular bisector of By the Incenter Theorem, since G is equidistant from

the sides of , EG = FG. Find FG using the Pythagorean Theorem. Therefore, .

ANSWER: 9

Since length cannot be negative, use only the positive 13. XZ square root, 5. Since EG = FG, EG = 5.

ANSWER: 5

Find each measure. SOLUTION: 12. RS From the figure, Thus,

Substitute y = 8 in XZ.

SOLUTION: Here RT = TS. By the converse of the Perpendicular Bisector Theorem, is a perpendicular bisector of

Therefore, . ANSWER: 34 14. BASEBALL Jackson, Trevor, and Scott are warming up before a baseball game. One of their warm-up drills requires three players to form a ANSWER: triangle, with one player in the middle. Where should 9 the fourth player stand so that he is the same distance from the other three players? 13. XZ

SOLUTION: SOLUTION: From the figure, The players can be represented by the vertices of a Thus, triangle. The point that is equidistant from each vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular

Substitute y = 8 in XZ. bisector of each side of the triangle.

ANSWER: 34

14. BASEBALL Jackson, Trevor, and Scott are ANSWER: warming up before a baseball game. One of their warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players?

15. The vertices of are D(0, 0), E(0, 7), and F(6, SOLUTION: 3). Find the coordinates of the orthocenter of . The players can be represented by the vertices of a SOLUTION: triangle. The point that is equidistant from each vertex is called the circumcenter. Find the The slope of is or So, the slope of circumcenter by constructing the perpendicular bisector of each side of the triangle. the altitude, which is perpendicular to is .

Now, the equation of the altitude from D to is:

In the same way, we can find the equation of the altitude from E to ANSWER: The slope of is . So, the slope of the altitude, which is perpendicular to is –2.

The equation of the altitude is

Solve the equations to find the intersection point of the altitudes.

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

The slope of is or So, the slope of Substitute the value of x in one of the equations to find the y-coordinate. the altitude, which is perpendicular to is .

Now, the equation of the altitude from D to is:

So, the coordinates of the orthocenter of is .

In the same way, we can find the equation of the altitude from E to The slope of is . So, the slope of the altitude, which is perpendicular to is –2.

The equation of the altitude is

Solve the equations to find the intersection point of ANSWER: the altitudes. (2, 3)

16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates Substitute the value of x in one of the equations to of the point where the chain should attach to the find the y-coordinate. triangle?

SOLUTION: In order for the triangles to hang so that they are balanced parallel to the floor, each triangle must be attached to its chain at its centroid. This point is So, the coordinates of the orthocenter of is located at the intersection of the medians of the . triangle.

The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has ANSWER: endpoints at (0, 4) and (4.5, 4). An equation of the (2, 3) line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s 16. PROM Georgia is on the prom committee. She centroid, is the point (3, 4). wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates of the point where the chain should attach to the triangle? SOLUTION: In order for the triangles to hang so that they are balanced parallel to the floor, each triangle must be attached to its chain at its centroid. This point is ANSWER: located at the intersection of the medians of the (3, 4) triangle. List the angles and sides of each triangle in The midpoint of the side from (0, 4) to (6, 0) is order from smallest to largest. or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) 17. and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has SOLUTION: endpoints at (0, 4) and (4.5, 4). An equation of the The sides from shortest to longest are . line containing this median is y = 4. The intersection The angles opposite these sides are ∠S, ∠R, and of x = 3 and y = 4, and the location of the traingle’s ∠T, respectively. So the angles from smallest to centroid, is the point (3, 4). largest are ∠S, ∠R, and ∠T. ANSWER:

ANSWER: 18. (3, 4) SOLUTION: List the angles and sides of each triangle in Use the Triangle Angle-Sum Theorem to find the order from smallest to largest. angle measures of each angle in the triangle.

17. SOLUTION: The sides from shortest to longest are . Replace x with 5.6 to find angle measures. The angles opposite these sides are ∠S, ∠R, and ∠T, respectively. So the angles from smallest to largest are ∠S, ∠R, and ∠T.

ANSWER:

The angles from smallest to largest are ∠N, ∠L, 18. ∠M. The sides opposite these angles are SOLUTION: , respectively. So, the sides Use the Triangle Angle-Sum Theorem to find the from shortest to longest are . angle measures of each angle in the triangle. ANSWER: ∠N, ∠L, ∠M; , ,

SOLUTION: The girls' houses can be represented by the vertices

of a triangle. List the sides of the triangle in order The angles from smallest to largest are N, L, ∠ ∠ from shortest to longest. First find the missing angle ∠M. The sides opposite these angles are measure using the Triangle Angle-Sum Theorem. , respectively. So, the sides from shortest to longest are . m∠Irene = 180 – (37 + 53) or 90

ANSWER: So, the angles from smallest to largest are ∠Anna,

∠N, ∠L, ∠M; , , ∠Sarah, ∠Irene. The sides opposite these angles are the path from Sarah to Irene, the path from Irene to NEIGHBORHOODS 19. Anna, Sarah, and Irene live Anna, and the path from Sarah to Anna, respectively. at the intersections of the three roads that make the So, the shorter path is for Sarah to get Irene and then triangle shown. If the girls want to spend the go to Anna’s house. afternoon together, is it a shorter path for Anna to stop and get Sarah and go on to Irene’s house, or for ANSWER: Sarah to stop and get Irene and then go on to Anna’s The shorter path is for Sarah to get Irene and then go house? to Anna’s house.

State the assumption you would make to start an indirect proof of each statement. 20. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false.

ANSWER: SOLUTION: m∠A < m∠B The girls' houses can be represented by the vertices of a triangle. List the sides of the triangle in order from shortest to longest. First find the missing angle 21. measure using the Triangle Angle-Sum Theorem. SOLUTION:

To start an indirect proof, first assume that what you

m∠Irene = 180 – (37 + 53) or 90 are trying to prove is false. is not congruent

to . So, the angles from smallest to largest are ∠Anna, ∠Sarah, ∠Irene. The sides opposite these angles are ANSWER: the path from Sarah to Irene, the path from Irene to is not congruent to . Anna, and the path from Sarah to Anna, respectively. So, the shorter path is for Sarah to get Irene and then 22. is a right triangle. go to Anna’s house. SOLUTION: ANSWER: To start an indirect proof, first assume that what you The shorter path is for Sarah to get Irene and then go are trying to prove is false. is not a right to Anna’s house. triangle.

State the assumption you would make to start ANSWER: an indirect proof of each statement. is not a right triangle. 20. 23. If 3y < 12, then y < 4. SOLUTION: To start an indirect proof, first assume that what you SOLUTION: are trying to prove is false. To start an indirect proof, first assume that what you are trying to prove is false. If 3y < 12, then y ≥ 4. ANSWER: ANSWER: Chapterm A 5 < Study m B Guide and Review ∠ ∠

21. 24. Write an indirect proof to show that if two angles are complementary, neither angle is a right angle. SOLUTION: To start an indirect proof, first assume that what you SOLUTION: are trying to prove is false. is not congruent To start an indirect proof, first assume that what you to . are trying to prove is false. In this case, try to find a contradiction if you assume that x or y are right ANSWER: angles.

is not congruent to . Let the measure of one angle be x and the measure of the other angle be y. By the definition of 22. is a right triangle. complementary angles, x + y = 90. SOLUTION: To start an indirect proof, first assume that what you Step 1 Assume that the angle with the measure x is a are trying to prove is false. is not a right right angle. Then x = 90. triangle. Step 2 Since x = 90, then x + y > 90. This is a ANSWER: contradiction because we know that x + y = 90.

is not a right triangle. Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must 23. If 3y < 12, then y < 4. be false. Therefore, the conclusion that neither angle SOLUTION: is a right angle must be true. To start an indirect proof, first assume that what you ANSWER: are trying to prove is false. If 3y < 12, then y ≥ 4. Let the measure of one angle be x and the measure ANSWER: of the other angle be y. By the definition of

complementary angles, x + y = 90. 24. Write an indirect proof to show that if two angles are Step 1 Assume that the angle with the measure x is a complementary, neither angle is a right angle. right angle. Then x = 90. SOLUTION: Step 2 Since x = 90, then x + y > 90. This is a To start an indirect proof, first assume that what you contradiction because we know that x + y = 90. are trying to prove is false. In this case, try to find a contradiction if you assume that x or y are right Step 3 Since the assumption that one angle is a right angles. angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle Let the measure of one angle be x and the measure is a right angle must be true. of the other angle be y. By the definition of complementary angles, x + y = 90. 25. CONCESSIONS Isaac purchased two items at the Step 1 Assume that the angle with the measure x is a concession stand at the Houston Dynamo game and right angle. Then x = 90. spent over $10. Use indirect reasoning to show that at least one of the items he purchased was over $5. Step 2 Since x = 90, then x + y > 90. This is a SOLUTION:

contradiction because we know that x + y = 90. To start an indirect proof, first assume that what you

are trying to prove is false. In this case, try to find a Step 3 Since the assumption that one angle is a right contradiction if you assume that the cost of item x angle leads to a contradiction, the assumption must and the cost of item y are less than or equal to $5. be false. Therefore, the conclusion that neither angle is a right angle must be true. Let the cost of one item be x, and the cost of the eSolutionsANSWER: Manual - Powered by Cognero other item be y. Page 6 Given: x + y > 10 Let the measure of one angle be x and the measure Prove: x > 5 or y > 5 of the other angle be y. By the definition of Indirect Proof: complementary angles, x + y = 90. Step 1 Assume that and . Step 1 Assume that the angle with the measure x is a Step 2 If and , then or

right angle. Then x = 90. . This is a contradiction because we Step 2 Since x = 90, then x + y > 90. This is a know that x + y > 50. contradiction because we know that x + y = 90. Step 3 Since the assumption that and Step 3 Since the assumption that one angle is a right leads to a contradiction of a known fact, the angle leads to a contradiction, the assumption must assumption must be false. Therefore, the conclusion be false. Therefore, the conclusion that neither angle that x > 5 or y > 5 must be true. Thus, at least one item had to be over $5. is a right angle must be true. ANSWER: 25. CONCESSIONS Isaac purchased two items at the Let the cost of one item be x, and the cost of the concession stand at the Houston Dynamo game and spent over $10. Use indirect reasoning to show that other item be y. at least one of the items he purchased was over $5. Given: x + y > 10 SOLUTION: Prove: x > 5 or y > 5 To start an indirect proof, first assume that what you Indirect Proof: are trying to prove is false. In this case, try to find a Step 1 contradiction if you assume that the cost of item x Assume that and . and the cost of item y are less than or equal to $5. Step 2 If and , then , or . This is a contradiction because we Let the cost of one item be x, and the cost of the know that x + y > 10. other item be y. Given: x + y > 10 Step 3 Since the assumption that x ≤ 5 and Prove: x > 5 or y > 5 leads to a contradiction of a known fact, the Indirect Proof: assumption must be false. Therefore, the conclusion Step 1 Assume that and . that x > 5 or y > 5 must be true. Thus, at least one Step 2 If and , then or item had to be over $5. . This is a contradiction because we know that x + y > 50. Is it possible to form a triangle with the given lengths? If not, explain why not. Step 3 Since the assumption that and 26. 5, 6, 9 leads to a contradiction of a known fact, the SOLUTION: assumption must be false. Therefore, the conclusion Check each inequality. that x > 5 or y > 5 must be true. Thus, at least one 5 + 6 > 9

item had to be over $5. 5 + 9 > 6 ANSWER: 6 + 9 > 5 Since the sum of each pair of side lengths is greater Let the cost of one item be x, and the cost of the than the third side length, lengths of 5, 6, and 9 units other item be y. will form a triangle. Given: x + y > 10 ANSWER: Prove: x > 5 or y > 5 Yes Indirect Proof: Step 1 Assume that and . 27. 3, 4, 8 Step 2 If and , then , or SOLUTION: . This is a contradiction because we 3 + 4 < 8 know that x + y > 10. Since the sum of one pair of side lengths is not greater than the third side length, lengths 3, 4, and 8 Step 3 Since the assumption that x ≤ 5 and units will not form a triangle. leads to a contradiction of a known fact, the assumption must be false. Therefore, the conclusion ANSWER: that x > 5 or y > 5 must be true. Thus, at least one No; 3 + 4 < 8 item had to be over $5. Find the range for the measure of the third side of a triangle given the measure of two sides. Is it possible to form a triangle with the given 28. 5 ft, 7 ft lengths? If not, explain why not. 26. 5, 6, 9 SOLUTION: Let n represent the length of the third side. SOLUTION: Check each inequality. According to the Triangle Inequality Theorem, the 5 + 6 > 9 largest side cannot be greater than the sum of the 5 + 9 > 6 other two sides. 6 + 9 > 5 Since the sum of each pair of side lengths is greater If n is the largest side, then n must be less than 5 + than the third side length, lengths of 5, 6, and 9 units 7. Therefore, n < 12. will form a triangle. If n is not the largest side, then 7 is the largest and 7 ANSWER: must be less than 5 + n. Therefore, 2 < n.

Yes Combining these two inequalities, we get 2 < n < 12.

27. 3, 4, 8 ANSWER: Let x be the length of the third side. 2 ft < x < 12 ft SOLUTION: 3 + 4 < 8 Since the sum of one pair of side lengths is not 29. 10.5 cm, 4 cm greater than the third side length, lengths 3, 4, and 8 SOLUTION: units will not form a triangle. Let n represent the length of the third side. ANSWER: No; 3 + 4 < 8 According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the

Find the range for the measure of the third side other two sides. of a triangle given the measure of two sides. 28. 5 ft, 7 ft If n is the largest side, then n must be less than 10.5 + 4. Therefore, n < 14.5. SOLUTION: If n is not the largest side, then 10.5 is the largest and Let n represent the length of the third side. 10.5 must be less than 4 + n. Therefore, 6.5 < n.

According to the Triangle Inequality Theorem, the Combining these two inequalities, we get 6.5 < n < largest side cannot be greater than the sum of the 14.5. other two sides. ANSWER: If n is the largest side, then n must be less than 5 + Let x be the length of the third side. 6.5 cm < x < 7. Therefore, n < 12. 14.5 cm. If n is not the largest side, then 7 is the largest and 7

must be less than 5 + n. Therefore, 2 < n. 30. BIKES Leonard rides his bike to visit Josh. Since High Street is closed, he has to travel 2 miles down Combining these two inequalities, we get 2 < n < 12. Main Street and turn to travel 3 miles farther on 5th Street. If the three streets form a triangle with ANSWER: Leonard and Josh’s house as two of the vertices, Let x be the length of the third side. 2 ft < x < 12 ft find the range of the possible distance between Leonard and Josh’s houses when traveling straight

29. 10.5 cm, 4 cm down High Street. SOLUTION: SOLUTION: Let n represent the length of the third side. Let x be the distance between Leonard and Josh’s

houses when traveling straight down High Street.

According to the Triangle Inequality Theorem, the Next, set up and solve each of the three triangle largest side cannot be greater than the sum of the inequalities. other two sides. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1. If n is the largest side, then n must be less than 10.5 + 4. Therefore, n < 14.5. Notice that x > –1 is always true for any whole If n is not the largest side, then 10.5 is the largest and number measure for x. Combining the two remaining 10.5 must be less than 4 + n. Therefore, 6.5 < n. inequalities, the range of values that fit both inequalities is x > 1 and x < 5, which can be written Combining these two inequalities, we get 6.5 < n < as 1 mile < x < 5 miles. Therefore, the distance is 14.5. greater than 1 mile and less than 5 miles.

ANSWER: ANSWER: Let x be the length of the third side. 6.5 cm < x < The distance is greater than 1 mile and less than 5 14.5 cm. miles.

30. BIKES Leonard rides his bike to visit Josh. Since Compare the given measures. High Street is closed, he has to travel 2 miles down 31. m∠ABC, m∠DEF Main Street and turn to travel 3 miles farther on 5th Street. If the three streets form a triangle with Leonard and Josh’s house as two of the vertices, find the range of the possible distance between Leonard and Josh’s houses when traveling straight down High Street. SOLUTION:

Let x be the distance between Leonard and Josh’s houses when traveling straight down High Street. SOLUTION: In and , and Next, set up and solve each of the three triangle AC > DF. By the Converse of the Hinge Theorem, inequalities. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1. ANSWER:

m∠ABC > m∠DEF Notice that x > –1 is always true for any whole number measure for x. Combining the two remaining inequalities, the range of values that fit both 32. QT and RS inequalities is x > 1 and x < 5, which can be written as 1 mile < x < 5 miles. Therefore, the distance is greater than 1 mile and less than 5 miles.

ANSWER: The distance is greater than 1 mile and less than 5 miles.

Compare the given measures. 31. m∠ABC, m∠DEF

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: SOLUTION: QT > RS In and , and AC > DF. By the Converse of the Hinge Theorem, 33. BOATING Rose and Connor each row across a pond heading to the same point. Neither of them has rowed a boat before, so they both go off course as ANSWER: shown in the diagram. After two minutes, they have each traveled 50 yards. Who is closer to their m∠ABC > m∠DEF destination?

32. QT and RS

SOLUTION:

As indicated, the distance from the anchor icon to SOLUTION: each boat is congruent and the distanced from the In and , anchor to the destination point (the picnic table icon) and . is also congruent. We know that Connor's angle is larger than Rose's so, based on the Hinge Theorem, By the Hinge Theorem, . the distance that Connor has to travel to get to their ANSWER: destination point is further than Rose's. Therefore, Rose is closer to the destination. QT > RS ANSWER: 33. BOATING Rose and Connor each row across a Rose pond heading to the same point. Neither of them has rowed a boat before, so they both go off course as shown in the diagram. After two minutes, they have each traveled 50 yards. Who is closer to their destination?

ANSWER: false; orthocenter

ANSWER: false; medians 5. The perpendicular bisectors of a triangle are State whether each sentence is true or false . If concurrent lines. false, replace the underlined term to make a true sentence. SOLUTION: 1. The altitudes of a triangle intersect at the centroid. The perpendicular bisectors of a triangle are concurrent lines. The statement is true. SOLUTION: The centroid is the the point where the medians ANSWER: intersect. The orthocenter is the point where the true altitudes intersect. false; orthocenter 6. A proof by contradiction uses indirect reasoning. ANSWER: false; orthocenter SOLUTION: Indirect reasoning is key when writing a proof by contradiction. The statement is true. 2. The point of concurrency of the medians of a triangle is called the incenter. ANSWER: SOLUTION: true The point where the medians intersect is the 7. A median of a triangle connects the midpoint of one centroid. The point of concurrency of the angle side of the triangle to the midpoint of another side of bisectors of a triangleis called the incenter. The the triangle. sentence is false. "The point of concurrency of the angle bisectors of a triangle is called the incenter." is SOLUTION: the true sentence. A median of a triangle connects the vertex to the midpoint of the side opposite it. The sentence is false. ANSWER: The true sentence is "A median of a triangle false; angle bisectors connects the midpoint of one side of the triangle to the vertex opposite that side." 3. The circumcenter of a triangle is equidistant from the vertices of the triangle. ANSWER: SOLUTION: false; the vertex opposite that side The point that is equidistant from the vertices of a 8. The incenter is the point at which the angle bisectors triangle is called the circumcenter. The statement is of a triangle intersect. true. SOLUTION: ANSWER: The point where the angle bisectors intersect is true called the incenter. The statement is true.

4. To find the centroid of a triangle, first construct the ANSWER: angle bisectors. true SOLUTION: 9. Explain how to write a proof by contradiction. To find the centroid of a triangle, first construct the medians. The sentence is false. The true sentence is SOLUTION: "To find the centroid of a triangle, first construct the Assume that the conclusion is false and show that medians." this assumption leads to a statement that cannot be true. ANSWER: false; medians ANSWER: Assume that the conclusion is false and show that 5. The perpendicular bisectors of a triangle are this assumption leads to a statement that cannot be concurrent lines. true. SOLUTION: 10. Explain how to locate the largest angle in a scalene The perpendicular bisectors of a triangle are triangle. Then explain when a triangle does not have concurrent lines. The statement is true. one largest angle. ANSWER: SOLUTION: true The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be 6. A proof by contradiction uses indirect reasoning. two congruent angles that are larger than the third SOLUTION: angle, so the sides opposite the congruent angles are Indirect reasoning is key when writing a proof by longer than the base. In an equilateral triangle, all contradiction. The statement is true. angles are the same size.

ANSWER: false; the vertex opposite that side SOLUTION: By the Incenter Theorem, since G is equidistant from 8. The incenter is the point at which the angle bisectors the sides of , EG = FG. Find FG using the of a triangle intersect. Pythagorean Theorem. SOLUTION: The point where the angle bisectors intersect is called the incenter. The statement is true.

ANSWER: true 9. Explain how to write a proof by contradiction. Since length cannot be negative, use only the positive SOLUTION: square root, 5. Assume that the conclusion is false and show that Since EG = FG, EG = 5. this assumption leads to a statement that cannot be true. ANSWER: 5 ANSWER: Assume that the conclusion is false and show that Find each measure. this assumption leads to a statement that cannot be 12. RS true.

10. Explain how to locate the largest angle in a scalene triangle. Then explain when a triangle does not have one largest angle.

SOLUTION: SOLUTION: The largest angle in a scalene triangle is opposite the Here RT = TS. By the converse of the Perpendicular longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third Bisector Theorem, is a perpendicular bisector of angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all Therefore, . angles are the same size.

ANSWER: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third ANSWER: angle, so the sides opposite the congruent angles are 9 longer than the base. In an equilateral triangle, all angles are the same size. 13. XZ 11. Find EG if G is the incenter of .

SOLUTION: SOLUTION: From the figure, By the Incenter Theorem, since G is equidistant from Thus, the sides of , EG = FG. Find FG using the Pythagorean Theorem.

Substitute y = 8 in XZ.

ANSWER: Since length cannot be negative, use only the positive square root, 5. 34 Since EG = FG, EG = 5. 14. BASEBALL Jackson, Trevor, and Scott are ANSWER: warming up before a baseball game. One of their 5 warm-up drills requires three players to form a triangle, with one player in the middle. Where should Find each measure. the fourth player stand so that he is the same 12. RS distance from the other three players?

SOLUTION: Here RT = TS. By the converse of the Perpendicular SOLUTION: Bisector Theorem, is a perpendicular bisector of The players can be represented by the vertices of a triangle. The point that is equidistant from each Therefore, . vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular bisector of each side of the triangle.

ANSWER: 9

13. XZ

ANSWER:

SOLUTION: From the figure, Thus,

Substitute y = 8 in XZ.

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

ANSWER: The slope of is or So, the slope of 34 the altitude, which is perpendicular to is . 14. BASEBALL Jackson, Trevor, and Scott are

warming up before a baseball game. One of their Now, the equation of the altitude from D to is: warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players?

In the same way, we can find the equation of the altitude from E to The slope of is . So, the slope of the altitude, which is perpendicular to is –2.

SOLUTION: The equation of the altitude is The players can be represented by the vertices of a triangle. The point that is equidistant from each vertex is called the circumcenter. Find the Solve the equations to find the intersection point of circumcenter by constructing the perpendicular the altitudes. bisector of each side of the triangle.

Substitute the value of x in one of the equations to find the y-coordinate.

ANSWER:

So, the coordinates of the orthocenter of is .

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION: The slope of is or So, the slope of ANSWER: (2, 3) the altitude, which is perpendicular to is . 16. PROM Georgia is on the prom committee. She Now, the equation of the altitude from D to is: wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates of the point where the chain should attach to the triangle? In the same way, we can find the equation of the SOLUTION: altitude from E to In order for the triangles to hang so that they are The slope of is . So, the slope of the balanced parallel to the floor, each triangle must be altitude, which is perpendicular to is –2. attached to its chain at its centroid. This point is located at the intersection of the medians of the

The equation of the altitude is triangle.

The midpoint of the side from (0, 4) to (6, 0) is

Solve the equations to find the intersection point of or (3, 2). The midpoint of the side the altitudes.

from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the Substitute the value of x in one of the equations to line containing this median is y = 4. The intersection find the y-coordinate. of x = 3 and y = 4, and the location of the traingle’s

centroid, is the point (3, 4).

So, the coordinates of the orthocenter of is .

ANSWER: (3, 4) List the angles and sides of each triangle in order from smallest to largest. ANSWER: (2, 3)

16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She

sketched out one triangle on a coordinate plane with 17. coordinates (0, 4), (3, 8), and (6, 0). If each triangle SOLUTION: is to be hung by one chain, what are the coordinates The sides from shortest to longest are . of the point where the chain should attach to the triangle? The angles opposite these sides are ∠S, ∠R, and ∠T, respectively. So the angles from smallest to SOLUTION: largest are ∠S, ∠R, and ∠T. In order for the triangles to hang so that they are balanced parallel to the floor, each triangle must be ANSWER: attached to its chain at its centroid. This point is located at the intersection of the medians of the triangle.

The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4). 18. SOLUTION: One median of this triangle has endpoints at (3, 8) Use the Triangle Angle-Sum Theorem to find the and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has angle measures of each angle in the triangle. endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4).

Replace x with 5.6 to find angle measures.

ANSWER: (3, 4) List the angles and sides of each triangle in order from smallest to largest.

The angles from smallest to largest are ∠N, ∠L, ∠M. The sides opposite these angles are , respectively. So, the sides 17. from shortest to longest are . SOLUTION: The sides from shortest to longest are . ANSWER:

The angles opposite these sides are ∠S, ∠R, and ∠N, ∠L, ∠M; , , ∠T, respectively. So the angles from smallest to 19. NEIGHBORHOODS Anna, Sarah, and Irene live largest are ∠S, ∠R, and ∠T. at the intersections of the three roads that make the triangle shown. If the girls want to spend the ANSWER: afternoon together, is it a shorter path for Anna to stop and get Sarah and go on to Irene’s house, or for Sarah to stop and get Irene and then go on to Anna’s house?

18. SOLUTION: Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle. SOLUTION: The girls' houses can be represented by the vertices of a triangle. List the sides of the triangle in order from shortest to longest. First find the missing angle measure using the Triangle Angle-Sum Theorem.

m∠Irene = 180 – (37 + 53) or 90

So, the angles from smallest to largest are ∠Anna, Replace x with 5.6 to find angle measures. ∠Sarah, ∠Irene. The sides opposite these angles are the path from Sarah to Irene, the path from Irene to Anna, and the path from Sarah to Anna, respectively. So, the shorter path is for Sarah to get Irene and then go to Anna’s house. ANSWER: The shorter path is for Sarah to get Irene and then go

to Anna’s house.

State the assumption you would make to start an indirect proof of each statement. 20. The angles from smallest to largest are ∠N, ∠L, ∠M. The sides opposite these angles are SOLUTION: , respectively. So, the sides To start an indirect proof, first assume that what you from shortest to longest are . are trying to prove is false.

ANSWER: ANSWER: ∠N, ∠L, ∠M; , , m∠A < m∠B

19. NEIGHBORHOODS Anna, Sarah, and Irene live 21. at the intersections of the three roads that make the triangle shown. If the girls want to spend the SOLUTION: afternoon together, is it a shorter path for Anna to To start an indirect proof, first assume that what you stop and get Sarah and go on to Irene’s house, or for are trying to prove is false. is not congruent Sarah to stop and get Irene and then go on to Anna’s to . house? ANSWER: is not congruent to .

22. is a right triangle. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. is not a right triangle.

SOLUTION: ANSWER: The girls' houses can be represented by the vertices is not a right triangle. of a triangle. List the sides of the triangle in order from shortest to longest. First find the missing angle 23. If 3y < 12, then y < 4. measure using the Triangle Angle-Sum Theorem. SOLUTION:

To start an indirect proof, first assume that what you m∠Irene = 180 – (37 + 53) or 90 are trying to prove is false. If 3y < 12, then y ≥ 4.

So, the angles from smallest to largest are ∠Anna, ANSWER: ∠Sarah, ∠Irene. The sides opposite these angles are the path from Sarah to Irene, the path from Irene to Anna, and the path from Sarah to Anna, respectively. 24. Write an indirect proof to show that if two angles are So, the shorter path is for Sarah to get Irene and then complementary, neither angle is a right angle. go to Anna’s house. SOLUTION: ANSWER: To start an indirect proof, first assume that what you The shorter path is for Sarah to get Irene and then go are trying to prove is false. In this case, try to find a contradiction if you assume that x or y are right to Anna’s house. angles.

State the assumption you would make to start Let the measure of one angle be x and the measure an indirect proof of each statement. of the other angle be y. By the definition of 20. complementary angles, x + y = 90. SOLUTION: Step 1 Assume that the angle with the measure x is a To start an indirect proof, first assume that what you right angle. Then x = 90. are trying to prove is false. ANSWER: Step 2 Since x = 90, then x + y > 90. This is a contradiction because we know that x + y = 90. m∠A < m∠B Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must 21. be false. Therefore, the conclusion that neither angle SOLUTION: is a right angle must be true. To start an indirect proof, first assume that what you ANSWER: are trying to prove is false. is not congruent to . Let the measure of one angle be x and the measure of the other angle be y. By the definition of ANSWER: complementary angles, x + y = 90. is not congruent to . Step 1 Assume that the angle with the measure x is a 22. is a right triangle. right angle. Then x = 90. SOLUTION: Step 2 Since x = 90, then x + y > 90. This is a To start an indirect proof, first assume that what you contradiction because we know that x + y = 90. are trying to prove is false. is not a right Step 3 Since the assumption that one angle is a right triangle. angle leads to a contradiction, the assumption must ANSWER: be false. Therefore, the conclusion that neither angle is not a right triangle. is a right angle must be true.

23. If 3y < 12, then y < 4. 25. CONCESSIONS Isaac purchased two items at the concession stand at the Houston Dynamo game and SOLUTION: spent over $10. Use indirect reasoning to show that To start an indirect proof, first assume that what you at least one of the items he purchased was over $5. are trying to prove is false. If 3y < 12, then y ≥ 4. SOLUTION: ANSWER: To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a contradiction if you assume that the cost of item x 24. Write an indirect proof to show that if two angles are and the cost of item y are less than or equal to $5. complementary, neither angle is a right angle. SOLUTION: Let the cost of one item be x, and the cost of the other item be y. To start an indirect proof, first assume that what you Given: x + y > 10 are trying to prove is false. In this case, try to find a Prove: x > 5 or y > 5 contradiction if you assume that x or y are right Indirect Proof: angles. Step 1 Assume that and .

Let the measure of one angle be x and the measure of the other angle be y. By the definition of Step 2 If and , then or complementary angles, x + y = 90. . This is a contradiction because we know that x + y > 50. Step 1 Assume that the angle with the measure x is a right angle. Then x = 90. Step 3 Since the assumption that and leads to a contradiction of a known fact, the Step 2 Since x = 90, then x + y > 90. This is a assumption must be false. Therefore, the conclusion contradiction because we know that x + y = 90. that x > 5 or y > 5 must be true. Thus, at least one item had to be over $5. Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must ANSWER: be false. Therefore, the conclusion that neither angle Let the cost of one item be x, and the cost of the

is a right angle must be true. other item be y. ANSWER: Given: x + y > 10 Let the measure of one angle be x and the measure Prove: x > 5 or y > 5 of the other angle be y. By the definition of Indirect Proof: complementary angles, x + y = 90. Step 1 Assume that and . Step 1 Assume that the angle with the measure x is a Step 2 If and , then , or right angle. Then x = 90. . This is a contradiction because we Step 2 Since x = 90, then x + y > 90. This is a know that x + y > 10. contradiction because we know that x + y = 90. Step 3 Since the assumption that x ≤ 5 and Step 3 Since the assumption that one angle is a right leads to a contradiction of a known fact, the angle leads to a contradiction, the assumption must assumption must be false. Therefore, the conclusion be false. Therefore, the conclusion that neither angle that x > 5 or y > 5 must be true. Thus, at least one Chapter 5 Study Guide and Review is a right angle must be true. item had to be over $5.

25. CONCESSIONS Isaac purchased two items at the Is it possible to form a triangle with the given concession stand at the Houston Dynamo game and lengths? If not, explain why not. spent over $10. Use indirect reasoning to show that 26. 5, 6, 9 at least one of the items he purchased was over $5. SOLUTION: SOLUTION: Check each inequality. To start an indirect proof, first assume that what you 5 + 6 > 9 are trying to prove is false. In this case, try to find a 5 + 9 > 6 contradiction if you assume that the cost of item x 6 + 9 > 5 and the cost of item y are less than or equal to $5. Since the sum of each pair of side lengths is greater than the third side length, lengths of 5, 6, and 9 units Let the cost of one item be x, and the cost of the will form a triangle. other item be y. ANSWER: Given: x + y > 10 Prove: x > 5 or y > 5 Yes Indirect Proof: Step 1 Assume that and . 27. 3, 4, 8

Step 2 If and , then or SOLUTION: . This is a contradiction because we 3 + 4 < 8 know that x + y > 50. Since the sum of one pair of side lengths is not greater than the third side length, lengths 3, 4, and 8 Step 3 Since the assumption that and units will not form a triangle. leads to a contradiction of a known fact, the ANSWER: assumption must be false. Therefore, the conclusion that x > 5 or y > 5 must be true. Thus, at least one No; 3 + 4 < 8 item had to be over $5. Find the range for the measure of the third side ANSWER: of a triangle given the measure of two sides. Let the cost of one item be x, and the cost of the 28. 5 ft, 7 ft other item be y. SOLUTION: Given: x + y > 10 Let n represent the length of the third side.

Prove: x > 5 or y > 5 According to the Triangle Inequality Theorem, the Indirect Proof: largest side cannot be greater than the sum of the Step 1 Assume that and . other two sides.

Step 2 If and , then , or If n is the largest side, then n must be less than 5 + . This is a contradiction because we 7. Therefore, n < 12. know that x + y > 10. If n is not the largest side, then 7 is the largest and 7 must be less than 5 + n. Therefore, 2 < n. Step 3 Since the assumption that x ≤ 5 and leads to a contradiction of a known fact, the Combining these two inequalities, we get 2 < n < 12. assumption must be false. Therefore, the conclusion ANSWER: that x > 5 or y > 5 must be true. Thus, at least one Let x be the length of the third side. 2 ft < x < 12 ft item had to be over $5.

Is it possible to form a triangle with the given 29. 10.5 cm, 4 cm lengths? If not, explain why not. SOLUTION: 26. 5, 6, 9 Let n represent the length of the third side. eSolutionsSOLUTION: Manual - Powered by Cognero Page 7 Check each inequality. According to the Triangle Inequality Theorem, the 5 + 6 > 9 largest side cannot be greater than the sum of the 5 + 9 > 6 other two sides. 6 + 9 > 5 Since the sum of each pair of side lengths is greater If n is the largest side, then n must be less than 10.5 than the third side length, lengths of 5, 6, and 9 units + 4. Therefore, n < 14.5. will form a triangle. If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n. ANSWER: Yes Combining these two inequalities, we get 6.5 < n < 14.5.

27. 3, 4, 8 ANSWER: SOLUTION: Let x be the length of the third side. 6.5 cm < x < 3 + 4 < 8 14.5 cm. Since the sum of one pair of side lengths is not greater than the third side length, lengths 3, 4, and 8 30. BIKES Leonard rides his bike to visit Josh. Since units will not form a triangle. High Street is closed, he has to travel 2 miles down Main Street and turn to travel 3 miles farther on 5th ANSWER: Street. If the three streets form a triangle with No; 3 + 4 < 8 Leonard and Josh’s house as two of the vertices, find the range of the possible distance between Find the range for the measure of the third side Leonard and Josh’s houses when traveling straight of a triangle given the measure of two sides. down High Street. 28. 5 ft, 7 ft SOLUTION: SOLUTION: Let x be the distance between Leonard and Josh’s Let n represent the length of the third side. houses when traveling straight down High Street.

According to the Triangle Inequality Theorem, the Next, set up and solve each of the three triangle largest side cannot be greater than the sum of the inequalities. other two sides. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1. If n is the largest side, then n must be less than 5 + 7. Therefore, n < 12. Notice that x > –1 is always true for any whole If n is not the largest side, then 7 is the largest and 7 number measure for x. Combining the two remaining must be less than 5 + n. Therefore, 2 < n. inequalities, the range of values that fit both inequalities is x > 1 and x < 5, which can be written Combining these two inequalities, we get 2 < n < 12. as 1 mile < x < 5 miles. Therefore, the distance is greater than 1 mile and less than 5 miles. ANSWER: ANSWER: Let x be the length of the third side. 2 ft < x < 12 ft The distance is greater than 1 mile and less than 5

29. 10.5 cm, 4 cm miles. SOLUTION: Compare the given measures. Let n represent the length of the third side. 31. m∠ABC, m∠DEF

According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the other two sides.

If n is the largest side, then n must be less than 10.5 + 4. Therefore, n < 14.5. If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n. SOLUTION: In and , and Combining these two inequalities, we get 6.5 < n < AC > DF. By the Converse of the Hinge Theorem, 14.5.

ANSWER: ANSWER: Let x be the length of the third side. 6.5 cm < x < m∠ABC > m∠DEF 14.5 cm.

Next, set up and solve each of the three triangle SOLUTION: inequalities. In and ,

2 + 3 > x, 2 + x > 3, and 3 + x > 2 and . That is, 5 > x, x > 1, and x > –1. . By the Hinge Theorem, Notice that x > –1 is always true for any whole ANSWER: number measure for x. Combining the two remaining inequalities, the range of values that fit both QT > RS inequalities is x > 1 and x < 5, which can be written as 1 mile < x < 5 miles. Therefore, the distance is 33. BOATING Rose and Connor each row across a greater than 1 mile and less than 5 miles. pond heading to the same point. Neither of them has rowed a boat before, so they both go off course as ANSWER: shown in the diagram. After two minutes, they have The distance is greater than 1 mile and less than 5 each traveled 50 yards. Who is closer to their destination? miles.

Compare the given measures. 31. m∠ABC, m∠DEF

SOLUTION: SOLUTION: In and , and As indicated, the distance from the anchor icon to AC > DF. By the Converse of the Hinge Theorem, each boat is congruent and the distanced from the anchor to the destination point (the picnic table icon) is also congruent. We know that Connor's angle is ANSWER: larger than Rose's so, based on the Hinge Theorem, m∠ABC > m∠DEF the distance that Connor has to travel to get to their destination point is further than Rose's. Therefore, Rose is closer to the destination. 32. QT and RS ANSWER: Rose

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: QT > RS

ANSWER: false; orthocenter

2. The point of concurrency of the medians of a triangle ANSWER: is called the incenter. true SOLUTION: 7. A median of a triangle connects the midpoint of one The point where the medians intersect is the side of the triangle to the midpoint of another side of centroid. The point of concurrency of the angle the triangle. bisectors of a triangleis called the incenter. The SOLUTION: sentence is false. "The point of concurrency of the angle bisectors of a triangle is called the incenter." is A median of a triangle connects the vertex to the the true sentence. midpoint of the side opposite it. The sentence is false. The true sentence is "A median of a triangle ANSWER: connects the midpoint of one side of the triangle to false; angle bisectors the vertex opposite that side." 3. The circumcenter of a triangle is equidistant from the ANSWER: vertices of the triangle. false; the vertex opposite that side SOLUTION: 8. The incenter is the point at which the angle bisectors The point that is equidistant from the vertices of a of a triangle intersect. triangle is called the circumcenter. The statement is SOLUTION: true. The point where the angle bisectors intersect is ANSWER: called the incenter. The statement is true. true ANSWER: 4. To find the centroid of a triangle, first construct the true angle bisectors. 9. Explain how to write a proof by contradiction. SOLUTION: SOLUTION: To find the centroid of a triangle, first construct the medians. The sentence is false. The true sentence is Assume that the conclusion is false and show that "To find the centroid of a triangle, first construct the this assumption leads to a statement that cannot be medians." true.

ANSWER: ANSWER: false; medians Assume that the conclusion is false and show that this assumption leads to a statement that cannot be 5. The perpendicular bisectors of a triangle are true. concurrent lines. 10. Explain how to locate the largest angle in a scalene SOLUTION: triangle. Then explain when a triangle does not have The perpendicular bisectors of a triangle are one largest angle. concurrent lines. The statement is true. SOLUTION: ANSWER: The largest angle in a scalene triangle is opposite the true longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third 6. A proof by contradiction uses indirect reasoning. angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all SOLUTION: angles are the same size. Indirect reasoning is key when writing a proof by contradiction. The statement is true. ANSWER: The largest angle in a scalene triangle is opposite the ANSWER: longest side. In an isosceles triangle, there may be true two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are 7. A median of a triangle connects the midpoint of one longer than the base. In an equilateral triangle, all side of the triangle to the midpoint of another side of angles are the same size. the triangle. SOLUTION: 11. Find EG if G is the incenter of . A median of a triangle connects the vertex to the midpoint of the side opposite it. The sentence is false. The true sentence is "A median of a triangle connects the midpoint of one side of the triangle to the vertex opposite that side." SOLUTION: ANSWER: By the Incenter Theorem, since G is equidistant from false; the vertex opposite that side the sides of , EG = FG. Find FG using the 8. The incenter is the point at which the angle bisectors Pythagorean Theorem. of a triangle intersect. SOLUTION: The point where the angle bisectors intersect is called the incenter. The statement is true.

ANSWER: true Since length cannot be negative, use only the positive 9. Explain how to write a proof by contradiction. square root, 5. Since EG = FG, EG = 5. SOLUTION: Assume that the conclusion is false and show that ANSWER: this assumption leads to a statement that cannot be 5 true. Find each measure. ANSWER: 12. RS Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true.

10. Explain how to locate the largest angle in a scalene triangle. Then explain when a triangle does not have one largest angle. SOLUTION: SOLUTION: Here RT = TS. By the converse of the Perpendicular The largest angle in a scalene triangle is opposite the Bisector Theorem, is a perpendicular bisector of longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are Therefore, . longer than the base. In an equilateral triangle, all angles are the same size.

11. Find EG if G is the incenter of .

SOLUTION: From the figure, SOLUTION: Thus, By the Incenter Theorem, since G is equidistant from the sides of , EG = FG. Find FG using the Pythagorean Theorem. Substitute y = 8 in XZ.

SOLUTION: SOLUTION: Here RT = TS. By the converse of the Perpendicular The players can be represented by the vertices of a triangle. The point that is equidistant from each Bisector Theorem, is a perpendicular bisector of vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular Therefore, . bisector of each side of the triangle.

ANSWER: 9

13. XZ ANSWER:

SOLUTION: From the figure, Thus,

Substitute y = 8 in XZ. 15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

The slope of is or So, the slope of

ANSWER: the altitude, which is perpendicular to is . 34 Now, the equation of the altitude from D to is: 14. BASEBALL Jackson, Trevor, and Scott are warming up before a baseball game. One of their warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players? In the same way, we can find the equation of the altitude from E to The slope of is . So, the slope of the altitude, which is perpendicular to is –2.

The equation of the altitude is

SOLUTION: The players can be represented by the vertices of a triangle. The point that is equidistant from each Solve the equations to find the intersection point of vertex is called the circumcenter. Find the the altitudes. circumcenter by constructing the perpendicular bisector of each side of the triangle.

Substitute the value of x in one of the equations to find the y-coordinate.

ANSWER: So, the coordinates of the orthocenter of is .

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION: ANSWER: (2, 3) The slope of is or So, the slope of 16. PROM Georgia is on the prom committee. She the altitude, which is perpendicular to is . wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She Now, the equation of the altitude from D to is: sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates of the point where the chain should attach to the triangle? SOLUTION: In the same way, we can find the equation of the In order for the triangles to hang so that they are altitude from E to balanced parallel to the floor, each triangle must be The slope of is . So, the slope of the attached to its chain at its centroid. This point is located at the intersection of the medians of the

altitude, which is perpendicular to is –2. triangle.

The equation of the altitude is The midpoint of the side from (0, 4) to (6, 0) is

or (3, 2). The midpoint of the side

Solve the equations to find the intersection point of the altitudes. from (3, 8) to (6, 0) is or (4.5, 4).

So, the coordinates of the orthocenter of is .

ANSWER: (3, 4) List the angles and sides of each triangle in order from smallest to largest.

ANSWER: (2, 3)

16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the 17. ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with SOLUTION: coordinates (0, 4), (3, 8), and (6, 0). If each triangle The sides from shortest to longest are . is to be hung by one chain, what are the coordinates The angles opposite these sides are ∠S, ∠R, and of the point where the chain should attach to the ∠T, respectively. So the angles from smallest to triangle? largest are ∠S, ∠R, and ∠T. SOLUTION: In order for the triangles to hang so that they are ANSWER: balanced parallel to the floor, each triangle must be attached to its chain at its centroid. This point is located at the intersection of the medians of the triangle.

The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

18. from (3, 8) to (6, 0) is or (4.5, 4). SOLUTION: Use the Triangle Angle-Sum Theorem to find the One median of this triangle has endpoints at (3, 8) angle measures of each angle in the triangle. and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4).

Replace x with 5.6 to find angle measures.

ANSWER: (3, 4)

List the angles and sides of each triangle in order from smallest to largest.

The angles from smallest to largest are ∠N, ∠L, ∠M. The sides opposite these angles are , respectively. So, the sides from shortest to longest are . 17. ANSWER: SOLUTION: ∠N, ∠L, ∠M; , , The sides from shortest to longest are . The angles opposite these sides are ∠S, ∠R, and 19. NEIGHBORHOODS Anna, Sarah, and Irene live ∠T, respectively. So the angles from smallest to at the intersections of the three roads that make the triangle shown. If the girls want to spend the largest are ∠S, ∠R, and ∠T. afternoon together, is it a shorter path for Anna to ANSWER: stop and get Sarah and go on to Irene’s house, or for Sarah to stop and get Irene and then go on to Anna’s

house?

m∠Irene = 180 – (37 + 53) or 90

ANSWER: The shorter path is for Sarah to get Irene and then go to Anna’s house.

State the assumption you would make to start an indirect proof of each statement. 20. SOLUTION: The angles from smallest to largest are ∠N, ∠L, To start an indirect proof, first assume that what you M. The sides opposite these angles are ∠ are trying to prove is false. , respectively. So, the sides from shortest to longest are . ANSWER: m∠A < m∠B ANSWER: ∠N, ∠L, ∠M; , , 21. 19. NEIGHBORHOODS Anna, Sarah, and Irene live SOLUTION: at the intersections of the three roads that make the triangle shown. If the girls want to spend the To start an indirect proof, first assume that what you afternoon together, is it a shorter path for Anna to are trying to prove is false. is not congruent stop and get Sarah and go on to Irene’s house, or for to . Sarah to stop and get Irene and then go on to Anna’s house? ANSWER: is not congruent to .

22. is a right triangle. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. is not a right triangle.

ANSWER:

is not a right triangle. SOLUTION: The girls' houses can be represented by the vertices 23. If 3y < 12, then y < 4. of a triangle. List the sides of the triangle in order from shortest to longest. First find the missing angle SOLUTION: measure using the Triangle Angle-Sum Theorem. To start an indirect proof, first assume that what you are trying to prove is false. If 3y < 12, then y ≥ 4. m Irene = 180 – (37 + 53) or 90 ∠ ANSWER:

So, the angles from smallest to largest are ∠Anna, ∠Sarah, ∠Irene. The sides opposite these angles are 24. Write an indirect proof to show that if two angles are the path from Sarah to Irene, the path from Irene to complementary, neither angle is a right angle. Anna, and the path from Sarah to Anna, respectively. So, the shorter path is for Sarah to get Irene and then SOLUTION: go to Anna’s house. To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a ANSWER: contradiction if you assume that x or y are right The shorter path is for Sarah to get Irene and then go angles. to Anna’s house. Let the measure of one angle be x and the measure State the assumption you would make to start of the other angle be y. By the definition of an indirect proof of each statement. complementary angles, x + y = 90. 20. Step 1 Assume that the angle with the measure x is a SOLUTION: right angle. Then x = 90. To start an indirect proof, first assume that what you are trying to prove is false. Step 2 Since x = 90, then x + y > 90. This is a contradiction because we know that x + y = 90. ANSWER: Step 3 Since the assumption that one angle is a right m∠A < m∠B angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle 21. is a right angle must be true. SOLUTION: ANSWER: To start an indirect proof, first assume that what you Let the measure of one angle be x and the measure are trying to prove is false. is not congruent of the other angle be y. By the definition of to . complementary angles, x + y = 90. ANSWER: Step 1 Assume that the angle with the measure x is a

is not congruent to . right angle. Then x = 90. 22. is a right triangle. Step 2 Since x = 90, then x + y > 90. This is a contradiction because we know that x + y = 90. SOLUTION: Step 3 Since the assumption that one angle is a right To start an indirect proof, first assume that what you are trying to prove is false. is not a right angle leads to a contradiction, the assumption must triangle. be false. Therefore, the conclusion that neither angle ANSWER: is a right angle must be true.

is not a right triangle. 25. CONCESSIONS Isaac purchased two items at the concession stand at the Houston Dynamo game and 23. If 3y < 12, then y < 4. spent over $10. Use indirect reasoning to show that SOLUTION: at least one of the items he purchased was over $5. To start an indirect proof, first assume that what you SOLUTION: are trying to prove is false. If 3y < 12, then y ≥ 4. To start an indirect proof, first assume that what you ANSWER: are trying to prove is false. In this case, try to find a contradiction if you assume that the cost of item x

and the cost of item y are less than or equal to $5.

24. Write an indirect proof to show that if two angles are complementary, neither angle is a right angle. Let the cost of one item be x, and the cost of the other item be y. SOLUTION: Given: x + y > 10 To start an indirect proof, first assume that what you Prove: x > 5 or y > 5 are trying to prove is false. In this case, try to find a Indirect Proof: contradiction if you assume that x or y are right Step 1 Assume that and . angles. Step 2 If and , then or Let the measure of one angle be x and the measure . This is a contradiction because we of the other angle be y. By the definition of know that x + y > 50. complementary angles, x + y = 90. Step 3 Since the assumption that and Step 1 Assume that the angle with the measure x is a leads to a contradiction of a known fact, the right angle. Then x = 90. assumption must be false. Therefore, the conclusion that x > 5 or y > 5 must be true. Thus, at least one Step 2 Since x = 90, then x + y > 90. This is a item had to be over $5. contradiction because we know that x + y = 90. ANSWER: Step 3 Since the assumption that one angle is a right Let the cost of one item be x, and the cost of the angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle other item be y. is a right angle must be true. Given: x + y > 10 ANSWER: Prove: x > 5 or y > 5 Let the measure of one angle be x and the measure Indirect Proof: of the other angle be y. By the definition of Step 1 Assume that and . complementary angles, x + y = 90. Step 2 If and , then , or Step 1 Assume that the angle with the measure x is a . This is a contradiction because we right angle. Then x = 90. know that x + y > 10. Step 2 Since x = 90, then x + y > 90. This is a Step 3 Since the assumption that x ≤ 5 and contradiction because we know that x + y = 90. leads to a contradiction of a known fact, the Step 3 Since the assumption that one angle is a right assumption must be false. Therefore, the conclusion angle leads to a contradiction, the assumption must that x > 5 or y > 5 must be true. Thus, at least one be false. Therefore, the conclusion that neither angle item had to be over $5.

is a right angle must be true. Is it possible to form a triangle with the given lengths? If not, explain why not. 25. CONCESSIONS Isaac purchased two items at the 26. 5, 6, 9 concession stand at the Houston Dynamo game and spent over $10. Use indirect reasoning to show that SOLUTION: at least one of the items he purchased was over $5. Check each inequality. 5 + 6 > 9 SOLUTION: 5 + 9 > 6 To start an indirect proof, first assume that what you 6 + 9 > 5 are trying to prove is false. In this case, try to find a Since the sum of each pair of side lengths is greater contradiction if you assume that the cost of item x than the third side length, lengths of 5, 6, and 9 units and the cost of item y are less than or equal to $5. will form a triangle.

Let the cost of one item be x, and the cost of the ANSWER: other item be y. Yes Given: x + y > 10 Prove: x > 5 or y > 5 Indirect Proof: 27. 3, 4, 8 Step 1 Assume that and . SOLUTION:

3 + 4 < 8 Step 2 If and , then or Since the sum of one pair of side lengths is not . This is a contradiction because we greater than the third side length, lengths 3, 4, and 8 know that x + y > 50. units will not form a triangle.

Given: x + y > 10 According to the Triangle Inequality Theorem, the Prove: x > 5 or y > 5 largest side cannot be greater than the sum of the other two sides. Indirect Proof:

Step 1 Assume that and . If n is the largest side, then n must be less than 5 + Step 2 If and , then , or 7. Therefore, n < 12. If n is not the largest side, then 7 is the largest and 7 . This is a contradiction because we must be less than 5 + n. Therefore, 2 < n. know that x + y > 10. Step 3 Since the assumption that x ≤ 5 and Combining these two inequalities, we get 2 < n < 12. leads to a contradiction of a known fact, the ANSWER: assumption must be false. Therefore, the conclusion Let x be the length of the third side. 2 ft < x < 12 ft that x > 5 or y > 5 must be true. Thus, at least one item had to be over $5. 29. 10.5 cm, 4 cm Is it possible to form a triangle with the given SOLUTION: lengths? If not, explain why not. Let n represent the length of the third side. 26. 5, 6, 9 According to the Triangle Inequality Theorem, the SOLUTION: largest side cannot be greater than the sum of the Check each inequality. other two sides. 5 + 6 > 9 5 + 9 > 6 If n is the largest side, then n must be less than 10.5 6 + 9 > 5 + 4. Therefore, n < 14.5. Since the sum of each pair of side lengths is greater If n is not the largest side, then 10.5 is the largest and than the third side length, lengths of 5, 6, and 9 units 10.5 must be less than 4 + n. Therefore, 6.5 < n. will form a triangle.

ANSWER: Combining these two inequalities, we get 6.5 < n < 14.5. Yes ANSWER: 27. 3, 4, 8 Let x be the length of the third side. 6.5 cm < x < SOLUTION: 14.5 cm. 3 + 4 < 8 Since the sum of one pair of side lengths is not 30. BIKES Leonard rides his bike to visit Josh. Since greater than the third side length, lengths 3, 4, and 8 High Street is closed, he has to travel 2 miles down units will not form a triangle. Main Street and turn to travel 3 miles farther on 5th Street. If the three streets form a triangle with ANSWER: Leonard and Josh’s house as two of the vertices, No; 3 + 4 < 8 find the range of the possible distance between Leonard and Josh’s houses when traveling straight

Find the range for the measure of the third side down High Street. of a triangle given the measure of two sides. SOLUTION: 28. 5 ft, 7 ft Let x be the distance between Leonard and Josh’s SOLUTION: houses when traveling straight down High Street.

Notice that x > –1 is always true for any whole If n is the largest side, then n must be less than 5 + number measure for x. Combining the two remaining 7. Therefore, n < 12. inequalities, the range of values that fit both If n is not the largest side, then 7 is the largest and 7 inequalities is x > 1 and x < 5, which can be written must be less than 5 + n. Therefore, 2 < n. as 1 mile < x < 5 miles. Therefore, the distance is greater than 1 mile and less than 5 miles. Combining these two inequalities, we get 2 < n < 12. ANSWER: ANSWER: The distance is greater than 1 mile and less than 5 ChapterLet x 5 be Study the length Guide of and the thirdReview side. 2 ft < x < 12 ft miles.

29. 10.5 cm, 4 cm Compare the given measures. 31. m ABC, m DEF SOLUTION: ∠ ∠ Let n represent the length of the third side.

According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the other two sides.

If n is the largest side, then n must be less than 10.5

+ 4. Therefore, n < 14.5. SOLUTION: If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n. In and , and AC > DF. By the Converse of the Hinge Theorem, Combining these two inequalities, we get 6.5 < n < 14.5. ANSWER: ANSWER: m∠ABC > m∠DEF Let x be the length of the third side. 6.5 cm < x < 14.5 cm. 32. QT and RS

Next, set up and solve each of the three triangle In and , inequalities. and . 2 + 3 > x, 2 + x > 3, and 3 + x > 2 By the Hinge Theorem, . That is, 5 > x, x > 1, and x > –1. ANSWER: Notice that x > –1 is always true for any whole QT > RS number measure for x. Combining the two remaining inequalities, the range of values that fit both 33. BOATING Rose and Connor each row across a inequalities is x > 1 and x < 5, which can be written pond heading to the same point. Neither of them has as 1 mile < x < 5 miles. Therefore, the distance is rowed a boat before, so they both go off course as greater than 1 mile and less than 5 miles. shown in the diagram. After two minutes, they have each traveled 50 yards. Who is closer to their ANSWER: destination? The distance is greater than 1 mile and less than 5 miles.

Compare the given measures. 31. m∠ABC, m∠DEF eSolutions Manual - Powered by Cognero Page 8

SOLUTION: SOLUTION: As indicated, the distance from the anchor icon to each boat is congruent and the distanced from the , In and and anchor to the destination point (the picnic table icon) AC > DF. By the Converse of the Hinge Theorem, is also congruent. We know that Connor's angle is larger than Rose's so, based on the Hinge Theorem, the distance that Connor has to travel to get to their ANSWER: destination point is further than Rose's. Therefore, m∠ABC > m∠DEF Rose is closer to the destination.

ANSWER: 32. QT and RS Rose

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: QT > RS

ANSWER: false; orthocenter

ANSWER: false; medians 5. The perpendicular bisectors of a triangle are concurrent lines. SOLUTION: The perpendicular bisectors of a triangle are concurrent lines. The statement is true.

ANSWER: true 6. A proof by contradiction uses indirect reasoning. SOLUTION: Indirect reasoning is key when writing a proof by contradiction. The statement is true.

ANSWER: true 7. A median of a triangle connects the midpoint of one side of the triangle to the midpoint of another side of the triangle. SOLUTION: A median of a triangle connects the vertex to the midpoint of the side opposite it. The sentence is false. The true sentence is "A median of a triangle connects the midpoint of one side of the triangle to the vertex opposite that side."

ANSWER: false; the vertex opposite that side 8. The incenter is the point at which the angle bisectors of a triangle intersect. SOLUTION: The point where the angle bisectors intersect is called the incenter. The statement is true.

ANSWER: true 9. Explain how to write a proof by contradiction. SOLUTION: Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true.

ANSWER: Assume that the conclusion is false and show that this assumption leads to a statement that cannot be true.

10. Explain how to locate the largest angle in a scalene triangle. Then explain when a triangle does not have one largest angle. SOLUTION: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all angles are the same size.

ANSWER: The largest angle in a scalene triangle is opposite the longest side. In an isosceles triangle, there may be two congruent angles that are larger than the third angle, so the sides opposite the congruent angles are longer than the base. In an equilateral triangle, all angles are the same size.

11. Find EG if G is the incenter of .

SOLUTION: By the Incenter Theorem, since G is equidistant from the sides of , EG = FG. Find FG using the Pythagorean Theorem.

Since length cannot be negative, use only the positive square root, 5. Since EG = FG, EG = 5.

ANSWER: 5 Find each measure. 12. RS

SOLUTION: Here RT = TS. By the converse of the Perpendicular Bisector Theorem, is a perpendicular bisector of

Therefore, .

ANSWER: 9

13. XZ

SOLUTION: From the figure, Thus,

Substitute y = 8 in XZ.

ANSWER: 34

14. BASEBALL Jackson, Trevor, and Scott are warming up before a baseball game. One of their warm-up drills requires three players to form a triangle, with one player in the middle. Where should the fourth player stand so that he is the same distance from the other three players?

SOLUTION: The players can be represented by the vertices of a triangle. The point that is equidistant from each vertex is called the circumcenter. Find the circumcenter by constructing the perpendicular bisector of each side of the triangle.

ANSWER:

15. The vertices of are D(0, 0), E(0, 7), and F(6, 3). Find the coordinates of the orthocenter of . SOLUTION:

The slope of is or So, the slope of

the altitude, which is perpendicular to is .

Now, the equation of the altitude from D to is:

In the same way, we can find the equation of the altitude from E to The slope of is . So, the slope of the altitude, which is perpendicular to is –2.

The equation of the altitude is

Solve the equations to find the intersection point of the altitudes.

Substitute the value of x in one of the equations to find the y-coordinate.

So, the coordinates of the orthocenter of is .

ANSWER: (2, 3)

16. PROM Georgia is on the prom committee. She wants to hang a dozen congruent triangles from the ceiling so that they are parallel to the floor. She sketched out one triangle on a coordinate plane with coordinates (0, 4), (3, 8), and (6, 0). If each triangle is to be hung by one chain, what are the coordinates of the point where the chain should attach to the triangle? SOLUTION: In order for the triangles to hang so that they are balanced parallel to the floor, each triangle must be attached to its chain at its centroid. This point is located at the intersection of the medians of the triangle.

The midpoint of the side from (0, 4) to (6, 0) is or (3, 2). The midpoint of the side

from (3, 8) to (6, 0) is or (4.5, 4).

One median of this triangle has endpoints at (3, 8) and (3, 2). An equation of the line containing this median is x = 3. Another median of this triangle has endpoints at (0, 4) and (4.5, 4). An equation of the line containing this median is y = 4. The intersection of x = 3 and y = 4, and the location of the traingle’s centroid, is the point (3, 4).

ANSWER: (3, 4) List the angles and sides of each triangle in order from smallest to largest.

17. SOLUTION: The sides from shortest to longest are . The angles opposite these sides are ∠S, ∠R, and ∠T, respectively. So the angles from smallest to largest are ∠S, ∠R, and ∠T. ANSWER:

18. SOLUTION: Use the Triangle Angle-Sum Theorem to find the angle measures of each angle in the triangle.

Replace x with 5.6 to find angle measures.

The angles from smallest to largest are ∠N, ∠L, ∠M. The sides opposite these angles are , respectively. So, the sides from shortest to longest are .

ANSWER: ∠N, ∠L, ∠M; , ,

m∠Irene = 180 – (37 + 53) or 90

ANSWER: The shorter path is for Sarah to get Irene and then go to Anna’s house.

State the assumption you would make to start an indirect proof of each statement. 20. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false.

ANSWER: m∠A < m∠B

21. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. is not congruent to .

ANSWER: is not congruent to .

22. is a right triangle. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. is not a right triangle.

ANSWER: is not a right triangle.

23. If 3y < 12, then y < 4. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. If 3y < 12, then y ≥ 4.

ANSWER:

24. Write an indirect proof to show that if two angles are complementary, neither angle is a right angle. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a contradiction if you assume that x or y are right angles.

Let the measure of one angle be x and the measure of the other angle be y. By the definition of complementary angles, x + y = 90.

Step 1 Assume that the angle with the measure x is a right angle. Then x = 90.

Step 2 Since x = 90, then x + y > 90. This is a contradiction because we know that x + y = 90.

Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle is a right angle must be true.

ANSWER: Let the measure of one angle be x and the measure of the other angle be y. By the definition of complementary angles, x + y = 90. Step 1 Assume that the angle with the measure x is a right angle. Then x = 90. Step 2 Since x = 90, then x + y > 90. This is a contradiction because we know that x + y = 90. Step 3 Since the assumption that one angle is a right angle leads to a contradiction, the assumption must be false. Therefore, the conclusion that neither angle is a right angle must be true.

25. CONCESSIONS Isaac purchased two items at the concession stand at the Houston Dynamo game and spent over $10. Use indirect reasoning to show that at least one of the items he purchased was over $5. SOLUTION: To start an indirect proof, first assume that what you are trying to prove is false. In this case, try to find a contradiction if you assume that the cost of item x and the cost of item y are less than or equal to $5.

Let the cost of one item be x, and the cost of the other item be y. Given: x + y > 10 Prove: x > 5 or y > 5 Indirect Proof: Step 1 Assume that and .

Step 2 If and , then or . This is a contradiction because we know that x + y > 50.

Step 3 Since the assumption that and leads to a contradiction of a known fact, the assumption must be false. Therefore, the conclusion that x > 5 or y > 5 must be true. Thus, at least one item had to be over $5.

ANSWER: Let the cost of one item be x, and the cost of the other item be y. Given: x + y > 10 Prove: x > 5 or y > 5 Indirect Proof: Step 1 Assume that and . Step 2 If and , then , or . This is a contradiction because we know that x + y > 10. Step 3 Since the assumption that x ≤ 5 and leads to a contradiction of a known fact, the assumption must be false. Therefore, the conclusion that x > 5 or y > 5 must be true. Thus, at least one item had to be over $5.

Is it possible to form a triangle with the given lengths? If not, explain why not. 26. 5, 6, 9 SOLUTION: Check each inequality. 5 + 6 > 9 5 + 9 > 6 6 + 9 > 5 Since the sum of each pair of side lengths is greater than the third side length, lengths of 5, 6, and 9 units will form a triangle.

ANSWER: Yes

27. 3, 4, 8 SOLUTION: 3 + 4 < 8 Since the sum of one pair of side lengths is not greater than the third side length, lengths 3, 4, and 8 units will not form a triangle.

ANSWER: No; 3 + 4 < 8

Find the range for the measure of the third side of a triangle given the measure of two sides. 28. 5 ft, 7 ft SOLUTION: Let n represent the length of the third side.

According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the other two sides.

If n is the largest side, then n must be less than 5 + 7. Therefore, n < 12. If n is not the largest side, then 7 is the largest and 7 must be less than 5 + n. Therefore, 2 < n.

Combining these two inequalities, we get 2 < n < 12.

ANSWER: Let x be the length of the third side. 2 ft < x < 12 ft

29. 10.5 cm, 4 cm SOLUTION: Let n represent the length of the third side.

According to the Triangle Inequality Theorem, the largest side cannot be greater than the sum of the other two sides.

If n is the largest side, then n must be less than 10.5 + 4. Therefore, n < 14.5. If n is not the largest side, then 10.5 is the largest and 10.5 must be less than 4 + n. Therefore, 6.5 < n.

Combining these two inequalities, we get 6.5 < n < 14.5.

ANSWER: Let x be the length of the third side. 6.5 cm < x < 14.5 cm.

Next, set up and solve each of the three triangle inequalities. 2 + 3 > x, 2 + x > 3, and 3 + x > 2 That is, 5 > x, x > 1, and x > –1.

Notice that x > –1 is always true for any whole number measure for x. Combining the two remaining inequalities, the range of values that fit both inequalities is x > 1 and x < 5, which can be written as 1 mile < x < 5 miles. Therefore, the distance is greater than 1 mile and less than 5 miles.

ANSWER: The distance is greater than 1 mile and less than 5 miles.

Compare the given measures. 31. m∠ABC, m∠DEF

SOLUTION: In and , and AC > DF. By the Converse of the Hinge Theorem,

ANSWER: m∠ABC > m∠DEF

32. QT and RS

SOLUTION: In and , and . By the Hinge Theorem, .

ANSWER: Chapter 5 Study Guide and Review QT > RS

ANSWER: Rose