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Park Forest Math Team

Meet #3 GeometryGeometry

Self-study Packet

Problem Categories for this Meet: 1. Mystery: Problem solving 2. : measures in plane figures including supplements and complements 3. Number Theory: Divisibility rules, factors, primes, composites 4. Arithmetic: Order of operations; mean, , mode; rounding; statistics 5. Algebra: Simplifying and evaluating expressions; solving equations with 1 unknown including identities

Important Information you need to know about GEOMETRY… Properties of ,

Formulas for Polygons where n means the number of sides:

• Exterior Angle Measurement of a Regular : 360÷n • Sum of Interior : 180(n – 2) • Interior Angle Measurement of a : • An interior angle and an exterior angle of a regular polygon always add up to 180°

Interior angle Exterior angle

Diagonals of a Polygon where n stands for the number of vertices (which is equal to the number of sides): • • A is a segment that connects one of a polygon to another vertex that is not directly next to it. The dashed lines represent some of the of this .

Pythagorean Theorem • a2 + b2 = c2 • a and b are the legs of the and c is the (the side opposite the ) c a

b

• Common Right are ones with sides 3, 4, 5, with sides 5, 12, 13, with sides 7, 24, 25, and multiples thereof—Memorize these!

Category 2 50th anniversary edition Geometry 26 Y Meet #3 January, 2014 W

1) How many cm long is segment 6 XY ? All measurements are in centimeters (cm). Z X 8

C 2) Angle ABC is a right X angle. Triangle BCD is E an D such that DB = BC. 120 150 Find the value of X if it is the measure in degrees of angle 70 EDC and X < 180. B A

3) Moe and Larry race from point B to point S at a rectangular . Moe runs from B to A to S at an average rate of 5 feet every second. Larry runs diagonally across the field from B to S at an average rate of 10 feet every 3 seconds. If they both leave point B at the same time, then who wins the race? Also, by how many seconds does the winner finish ahead of the runnerup? (You must answer both questions correctly to receive credit.)

E S ANSWERS

1) ______cm 50 feet

2) ______B 120 feet A 3) ______winner

______seconds

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Solutions to Category 2 Geometry Meet #3 January, 2014

Answers 1) Use the Pythagorean Theorem twice first to find the of WX and then XY. 1) 24

2) 155

3) Moe Use this result to find XY: 5

2) The measure of angle DBA is 20 degrees, because the sum of the angles of a is 360 degrees. The measure of angle DBC is 70 degrees, because angle ABC is a right angle (90 degrees). Since two sides (DB and BC) of triangle DBC are congruent, the angles opposite those sides are congruent. The vertex angle, DBC, measures 70 degrees, so the angles are 55 degrees each, including angle BDC. angle X + 150 + 55 = 360, so X = 155.

3) Both answers must be answered correctly in order for students to receive credit. Moe: runs 120 + 50, or 170 feet. At a rate of 5 feet per second, it takes him 170 / 5, or 34 seconds to reach point S. Larry: Use the Pythagorean Theorem to find that he has run 130 feet. At a rate of 10 feet every 3 seconds, it takes him (130 / 10) x 3, or 39 seconds to reach point S. Moe, therefore, reaches point S ahead of Larry by 39 34, or by 5 seconds, so Moe wins the race.

www.imlem.org Meet #3 January 2012

Category 2 – Geometry

(11, 4) 1. Given the coordinates in the diagram, what is the distance between the two points? (-1, -1)

2. How many diagonals are there in a regular polygon with sides (a )?

3. The sum of interior angles in a regular polygon is times as great as the measure of each of its exterior angles. How many sides does the polygon have?

Answers

1. ______Units

2. ______Diagonals

3. ______Sides

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Meet #3 January 2012

Solutions to Category 2 – Geometery Answers

1. 1. The horizontal distance is units, and the vertical distance is 2. units, so the total distance is √ units. 3.

2. The formula for the number of diagonals in a polygon with N sides is:

so in our case we’ll have diagonals.

3. The exterior angles of a polygon all add up to 360 degrees, so if there are

sides to the polygon, then each exterior angle measures degrees.

Every interior angle measures degrees, and their sum is therefore

degrees. So in our case we’re told that:

whice we can rewrite as: .

Though this is technically a , we know that is a natural number and can easily find that is a solution (an ). [The other solution, , is clearly not an answer to our problem].

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Category 2 - Geometry Meet #3, January 2010

1. The number of diagonals in a polygon is four times the number of its vertices. How many vertices does it have? (A diagonal is a segment that connects two non-adjacent vertices of a polygon).

2. The exterior angle to a regular polygon N (with N sides) is half that of a regular polygon M (with M sides). Polygon N has 7 times as many diagonals as polygon M. What is the value of 푀 ∙ 푁?

3. Tom stands exactly 2 miles west of Jerry. At 10:00am Tom starts walking east at 5 mph (miles per hour). At 10:20am Jerry starts heading north at 9 mph. How many miles between them at 12:00pm (noon)?

Answers 1. ______2. ______Remember: You do not have to specify units. Specifying 3. ______the wrong units will disqualify your answer. www.imlem.org

Solutions to Category 2 - Geometry Answers Meet #3, January 2010 1. 11 2. 50

3. 17

1. If we call the number of vertices V, then the number of diagonals is given by the 푉×(푉−3) 푉−3 formula: and for this to equal 4 × 푉 we get = 4 or 푉 = 11. 2 2

360 2. The exterior angle is so it should be clear that 푁 = 2푀. 푁푢푚푏푒푟 표푓 푠𝑖푑푒푠 푀∙(푀−3) 푁∙(푁−3) 2푀∙(2푀−3) Polygon M will have diagonals, and polygon N will have = 2 2 2 diagonals. So we need to solve: 2푀 ∙ 2 ∙ 푀 − 3 = 7 ∙ 푀 ∙ 푀 − 3 which we can simplify to 4 ∙ 푀 − 6 = 7푀 − 21 and then to 3 ∙ 푀 = 15 and the solution is 푀 = 5, 푁 = 10 and 푀 ∙ 푁 = 50.

3. At 12:00pm, having walked for two hours, Tom is 10 miles to the east of his original location, which means he’s 8 miles east of Jerry’s original location. Jerry, after running at 9 mph for 1 hour and 40 minutes is 15 miles north of his (own) original location. The distance between them is 82 + 152 = 289 = 17 푚𝑖푙푒푠.

Note that we have to compare their locations to some ‘fixed’ or agreed-upon point. You could have used any other point-of-reference and get the same answer.

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Category 2 Geometry Meet #3, January 2008

1. If the exterior angle of a regular polygon is 24 degrees, how many sides does the polygon have?

2. In the figure to the right, four of the sides of a regular polygon are shown. Also drawn are all of the diagonals that use A as one of the endpoints of the diagonal; however the full of the diagonals are not always shown. How many diagonals does this polygon have? A

3. In the figure below, triangle ABC is a with right angle at A and quadrilateral BCDE is a . If AB = 8 and AC = 18 , what is the of the pentagon ABEDC? B

C A

E

Answers

1. ______D 2. ______3. ______

Solutions to Category 2 Geometry Meet #3, January 2008

Answers 1. Since the sum of the exterior angles is always 360 degrees, if you divide 360 degrees by 24 degrees you get 15 which is the 1. 15 number of equal exterior angles and therefore 15 vertices and sides to the polygon. 2. 44

3. 20 2. If there are 8 diagonals and only 2 of the diagonals’ other endpoints are shown there must be 6 vertices that are not shown. Since 5 vertices are shown there are a total of 11

vertices and 11 sides. If there are 11 sides there are diagonals total.

3. Using the Pythagorean Theorem:

→ → Since is the area of the square, the area of square EBCD is 26. The pentagon we are looking for though is equal to the area of the square minus the area of the right triangle ABC. The area of triangle ABC = . The area of pentagon ABEDC = 26 – 6 = 20 .

B

C A

E

D Category 2 Geometry Meet #3, January 2006

1. How many degrees are in the measure of an interior angle of a regular 15-gon?

1 2. A certain polygon has 2 2 times as many diagonals as sides. How many sides are there on this polygon? Note: A diagonal is a that connects two non-adjacent vertices of a polygon.

3. Three right triangles are joined together to form the concave pentagon shown below. The measure of GA is 15 units, AS is 20 units, SM is 21 units, and ME is 35 units. How many units are in the of pentagon GAMES?

M

A E S Answers G 1. ______2. ______3. ______

www.imlem.org Solutions to Category 2 Geometry Meet #3, January 2006

Answers 1. If we draw line segments from one vertex to all the non-adjacent vertices, we can subdivide the 15-gon into 13 triangles. Each triangle has an angle sum of 180 1. 156 degrees, so the sum of the interior angles of the 15-gon must be 13 180 = 2340 degrees. Since the 15-gon is 2. 8 × regular, this total is shared equally among the 15 interior angles. Each interior angle must have a measure of 2340 3. 132 ÷ 15 = 156 degrees.

2. The polygon in question must have an even number of sides, since we are 1 multiplying by 2 2 and the number of diagonals must be a whole number. A square has only 2 diagonals, so let’s try a . From each vertex of a hexagon we can draw 3 diagonals. If we simply multiply 6 × 3, we will have counted each diagonal twice, so there must be only 6 × 3 ÷ 2 = 9 diagonals in a hexagon. This is 1 only 1 2 times the number of sides. Now let’s try an octagon. From each vertex of an octagon we can draw 5 diagonals. That’s 8 × 5 ÷ 2 = 20 diagonals in all. 1 Twenty is exactly 2 2 times 8, so our polygon has 8 sides.

3. We can use the Pythagorean Theorem to find the lengths of sides AM, GS, and SE. Let the the 2 2 2 measure of AM be x. Then 20 + 21 = x . This M means x 2 = 400 + 441 = 841. Since 292 = 841, x must be 29 units. Let the measure of GS be y. 29 35 2 2 2 21 Then 15 + 20 = y . (Here we might recognize 20 A E a multiple of the 3-4-5 .) This S 28 15 means y2 = 225 + 400 = 625. Since 252 = 625, 25 G y must be 25 units. Finally, let the measure of SE be z. Then 212 + z 2 = 35 2 . This means z 2 = 35 2 − 212 =1225 − 441 = 784. Since 282 = 784, z must be 28 units. The perimeter of polygon GAMES is thus 15 + 29 + 35 + 28 + 25 = 132 units.

www.imlem.org Category 2 Geometry Meet #3, January 2004

1. How many diagonals can be drawn in a ? Note: A decagon is a polygon with 10 sides and a diagonal of a polygon is a segment which connects any two non-consecutive vertices.

2. A regular polygon has an interior angle measure that is greater than 144 degrees and less than 150 degrees. How many sides does the polygon have?

3. In the figure at right, every quadrilateral is a square and every C triangle is a right triangle. The area of square ABCD is 144 square units and B F the area of square EFGH is 16 square E G units. Length LK is 5 units. How many H units are in the perimeter of the figure D (polygon ABCEFGIJKLMN)? I A K L J

Answers 1. ______2. ______N M 3. ______

www.Imlem.org Solutions to Category 2 Geometry Meet #3, January 2004

Answers 1. From each of the ten vertices of a decagon, one can draw seven diagonals. If we multiply 7 by 10 to get 70, 1. 35 we will have counted every diagonal twice. Therefore there must be 70 ÷ 2 = 35 diagonals in a decagon. 2. 11 2. A polygon with n sides can be subdivided into n – 2 3. 100 triangles, each with an angle sum of 180. This gives the n-gon a total angle sum of 180(n − 2) or 180n − 360. If the n-gon is regular, then all n of its interior angles will have the angle measure (180n − 360) n. We need to solve the double inequality 144 < (180n − 360) n <150 for a whole-number value of n. 144 < (180n − 360) n (180n − 360) n <150 144n <180n − 360 180n − 360 <150n 360 <180n −144n 180n −150n < 360 360 < 36n 30n < 360 10 < n n <12 The only whole number value between 10 and 12 is 11, so the polygon must have 11 sides.

3. Square ABCD has an area of 144 square units, so C its side length is 12. Length DL is 5, so triangle ADL 12 13 B 4 F 4 is a 5-12-13 triangle as the Pythagorean Theorem E G 2 2 2 confirms: 5 +12 = 25 +144 = 169 and 13 =169. D H 5 We are also given that square EFGH has an area of 16 12 I A K 3 square units, so its side length is 4 units. Triangle L 5 3 J 2 2 EHK is a 3-4-5 triangle, since 5 − 4 = 25 −16 = 9 13 13 and 32 = 9. We can now label all the side lengths in the figure as shown at right. The perimeter is N M 12 +12 + 13 + 4 + 4 + 5 +3 + 3 + 5 +13 +13 +13 = 100 13 units.

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