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2.1 Segment Bisectors

Goal Bisect a segment. Find Geo-Activity Folding a Segment Bisector the coordinates of the midpoint of a segment. ●1 On a piece of paper, draw AB&* . ●2 Fold the paper so that B is on top of point A. Key Words • midpoint • segment bisector A • bisect B B

●3 Label the point where the fold ●4 Use a ruler to measure the intersects AB&* as point M. lengths of AM&** , MB&* , and AB&* . What do you notice?

A A M M B B

In the Geo-Activity, M is called the midpoint of AB&* . The midpoint of a segment is the point on the segment that divides it into two congruent segments. Student Help A segment bisector is a segment, ray, line, or that intersects a segment at its midpoint. To bisect a segment means to divide the VOCABULARY TIP segment into two congruent segments. Bi- means “two,” and -sect means “to cut.” So, bisect means “to C cut in two.” M A M B A B D

M is the midpoint of AB&* . CD^&*( is a bisector of AB&* .

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Student Help EXAMPLE 1 Find Segment Lengths STUDY TIP M is the midpoint of AB&* . Find AM and MB . 26 The midpoint of a segment divides the A M B segment in half. Solution A M B M is the midpoint of AB&* , so AM and MB are each half the length of AB&* . If you know the length ᎏ1ᎏ ᎏ1ᎏ 1 AM ϭ MB ϭ p AB ϭ p 26 ϭ 13 of AB&* , multiply AB by ᎏᎏ 2 2 2 to find AM and MB . ANSWER ᮣ AM ϭ 13 and MB ϭ 13.

EXAMPLE 2 Find Segment Lengths P is the midpoint of RS&* . Find PS and RS . R 7 P S

Solution P is the midpoint of RS&* , so PS ϭ RP . Therefore, PS ϭ 7. You know that RS is twice the length of RP&* . RS ϭ 2 p RP ϭ 2 p 7 ϭ 14

ANSWER ᮣ PS ϭ 7 and RS ϭ 14.

Find Segment Lengths

1. Find DE and EF . 2. Find NP and MP .

18

D E F M 11 N P

EXAMPLE 3 Use Algebra with Segment Lengths Line l is a segment bisector of AB&* . l Find the value of x. A 5x M 35 B Solution AM ϭ MB Line l bisects AB&* at point M. 5x ϭ 35 Substitute 5 x for AM and 35 for MB . 5x 35 ᎏᎏ ϭ ᎏᎏ Divide each side by 5. 5 5 x ϭ 7 Simplify.

CHECK ✓Check your solution by substituting 7 for x. 5x ϭ 5( 7) ϭ 35

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Midpoints If you know the coordinates of the endpoints of a in a coordinate plane, you can find the coordinates of the midpoint of the segment using the Midpoint Formula.

THE MIDPOINT FORMULA

Words The coordinates of the y midpoint of a segment are the Student Help y2 B(x , y ) averages of the x-coordinates 2 2 READING TIP and the y-coordinates of the ϩ ϩ The numbers 1 and 2 in y ؉ y x1 x2 y1 y2 endpoints. 1 2 , M ͑ 2 2 ͒ x1 and y2 are called 2 The midpoint of the subscripts . You read x1 Symbols as “ x sub 1” and y2 as segment joining ”y sub 2.” y1 A(x1, y1) A(x1, y1) and B(x2, y2) ϩ ϩ x1 x2 y1 y2 is M ᎏᎏ, ᎏᎏ . x x ؉ x x ΂ 2 2 ΃ 1 1 2 2 x 2

EXAMPLE 4 Use the Midpoint Formula Find the coordinates of the midpoint of AB&* . a. A(1, 2), B(7, 4) b. A(Ϫ2, 3), B(5, Ϫ1) Solution Student Help First make a sketch. Then use the Midpoint Formula.

SKILLS REVIEW a. y b. y For help plotting points Ϫ in a coordinate plane, A( 2, 3) M see p. 664. B(7, 4) 1 M 1 x 1 A(1, 2) B(5, Ϫ1) 1 x

ϭ ϭ ؊ Let ( x1, y1) (1, 2) Let ( x1, y1) ( 2, 3) ϭ ϭ ؊ and ( x2, y2) (7, 4). and ( x2, y2) (5, 1). x ϩ x y ϩ y x ϩ x y ϩ y M ϭ ᎏ1 ᎏ2, ᎏ1 ᎏ2 M ϭ ᎏ1 ᎏ2, ᎏ1 ᎏ2 ΂ 2 2 ΃ ΂ 2 2 ΃ (ϩ 7 2 ϩ 4 ؊2 ϩ 5 3 ϩ (؊1 1 ϭ ᎏᎏ, ᎏᎏ ϭ ᎏᎏ, ᎏᎏ ΂ 2 2 ΃ ΂ 2 2 ΃ ϭ (4, 3) 3 ϭ ᎏᎏ, 1 ΂2 ΃

Use the Midpoint Formula

Sketch PQ&* . Then find the coordinates of its midpoint. 3. P(2, 5), Q(4, 3) 4. P(0, Ϫ2), Q(4, 0) 5. P(Ϫ1, 2), Q(Ϫ4, 1)

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2.12.1 Exercises Guided Practice

Vocabulary Check 1. In the diagram shown at the right, l name the midpoint and a segment bisector of AB&* . A C B

Skill Check M is the midpoint of the segment. Find the segment lengths. 2. Find RM and MS . 3. Find FM and MG .

6 50

R M S F M G 4. Find MQ and PQ . 5. Find YM and YZ .

P 4 M P Y M 20 Z

M is the midpoint of JK&* . Find the value of the variable. 6. 7. J x ؊ 14 M 33 K J 16 r M 48 K

Find the coordinates of the midpoint of PR&* .

8.y 9.y 10. y P (Ϫ2, 1) (5, 6) 1 R P(2, 3) 4 1 x (4, 1) 1 R P (Ϫ3, 0) 2 x 1 x R(1, Ϫ3)

Practice and Applications

Extra Practice Recognizing Midpoints In Exercises 11–14, determine whether M is See p. 677. the midpoint of AB&* . Explain your reasoning.

11. 12. A B M A M B

13. 14. A B A M M B 15. Visualize It! Sketch a line segment, PQ&* , that is bisected by line l at point R.

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Homework Help Finding Segment Lengths M is the midpoint of the segment. Find the segment lengths. Example 1: Exs. 16–19 16. 17. Example 2: Exs. 20–23 Find KM and ML . Find DM and ME. Example 3: Exs. 26–29 38 82 Example 4: Exs. 30–35 K M L D M E

18. Find YM and MZ . 19. Find AM and MB .

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Y M Z A M B

Finding Segment Lengths Line l bisects the segment. Find the segment lengths.

20. Find CB and AB . 21. Find MP and MN . l l

A 36 C B M P 15 N

22. Find FE and DE . 23. Find UT and ST . l l

D 29.5 F E S 3.6 U T

Biking The Minuteman Bikeway is a 10.5 mile bike path that runs from Arlington to Bedford, Massachusetts. 24. Caitlin and Laurie begin at opposite ends of the Minuteman Bikeway and meet at the halfway point on the path. How far does each rider bike?

25. Caitlin starts on the path 4.3 miles from the Arlington end. Laurie starts on the path 3 miles from the Bedford end. How far will each rider bike before reaching the halfway point on the path?

Using Algebra Find the value of the variable. 26. 27. A 6p M 72 B A 19 M q ؉ 7 B

28. 29. IStudent Help A r ؊ 3M 15 B A 4M 2 s ؉ 6 B ICLASSZONE.COM

HOMEWORK HELP Midpoint Formula Find the coordinates of the midpoint of PQ&* . Extra help with problem solving in Exs. 30–35 is 30. P(0, 0), Q(4, 6) 31. P(3, 8), Q(7, 6) 32. P(Ϫ5, 6), Q(9, 7) at classzone .com 33. P(Ϫ12, 0), Q(6, 1) 34. P(Ϫ4, 4), Q(4, 0) 35. P(3, 2), Q(Ϫ7, Ϫ4)

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Student Help Strike Zone In Exercises 36 and 37, use the information below. In baseball, the strike zone is the region a baseball needs to pass LOOK BACK through for the umpire to declare it a strike if the batter does not For more about baseball, see p. 50. swing. The top of the strike zone is a horizontal plane passing through the midpoint of the top of the batter’s shoulders and the top of the uniform pants when the player is in a batting stance. ᮣ Source: Major League Baseball 36. Find the coordinate of T. 37. Find the coordinate of T.

60 63 T T 45 42

24 22

0 0

Student Help EXAMPLE Latitude-Longitude Coordinates VOCABULARY TIP Find the coordinates of 122 °W 120 °W 118 °W 116 °W Lines of latitude run the place halfway between parallel to the Equator. ؇ ؇ NEVADA Lines of longitude run San Francisco ( 37.8 N, 122.4 W) ؇ ؇ north-south. and Los Angeles ( 34.1 N, 118.2 W). San Francisco (N (37.8 ؇N, 122.4 ؇W° 38 Solution CALIFORNIA x ϩ x y ϩ y M ϭ ᎏ1 ᎏ2, ᎏ1 ᎏ2 ΂ 2 2 ΃ 36 °N ؇ ϩ 34.1 ؇ 122.4 ؇ ϩ 118.2 ؇ 37.8 ϭ ᎏᎏ, ᎏᎏ ΂ 2 2 ΃ ° ϭ (35.95 Њ⌵ , 120.3 ЊW) 34 N Los Angeles (؇N, 118.2 ؇W 34.1)

ANSWER ᮣ The place halfway between San Francisco and Los Angeles has coordinates (35.95 ЊN, 120.3 ЊW).

Latitude-Longitude Coordinates Find the coordinates of the place halfway between the two cities in California. 38. Fresno: (36.7 ЊN, 119.8 ЊW) 39. Bishop: (37.4 ЊN, 118.4 ЊW) Napa: (38.3 ЊN, 122.3 ЊW) Los Angeles: (34.1 ЊN, 118.2 ЊW) 40. San Francisco: (37.8 ЊN, 122.4 ЊW) 41. Santa Barbara: (34.4 ЊN, 119.7 ЊW) Palo Alto: (37.4 ЊN, 122.1 ЊW) Oakland: (37.8 ЊN, 122.3 ЊW) 58 Chapter 2 Segments and Angles Page 7 of 7

42. Using Midpoints In the diagram below, B is the midpoint of AC&* , AB ϭ 9, and AD ϭ 25. Find CD .

25

A9 B C D

43. Challenge The midpoint of AB&* is M(7, 5). The coordinates of point A are (4, 1). Find the coordinates of point B. Explain.

Standardized Test 44. Multiple Choice T is the midpoint of QR&* . What is the value of x? Practice ɕA 17 ɕB 22 P 4x ؊ 10T 78 R ɕC 29.5 ɕD 88

45. Multiple Choice What is the midpoint of the segment joining (2, 7) and ( Ϫ6, 2)?

9 9 ɕF Ϫ2, ᎏᎏ ɕG (Ϫ4, 9) ɕH (Ϫ2, 4) ɕJ ᎏᎏ, Ϫ2 ΂ 2΃ ΂2 ΃

Mixed Review Evaluating Statements Use the diagram at the right to determine whether the statement is true or false . (Lessons 1.3, 1.5)

46. Point A lies on line m. l 47. l Point E lies on line . C m 48. Points B, E, and C are collinear. F B E A 49. Lines l and m intersect at point E. D 50. Point E is between points B and C. 51. Point F is between points A and B.

Classifying Angles Name the and sides of the angle. Then state whether it appears to be acute , right , obtuse , or straight . (Lesson 1.6) 52. 53. 54. P A J K L P R B C

Algebra Skills Evaluating Expressions Evaluate the expression. (Skills Review, p. 670) 1 55. 2 p 15 ϩ 40 56. 120 Ϫ 35 p 3 57. ᎏᎏ p 50 ϩ 145 2 5 58. ᎏᎏ p 16 Ϫ 20 59. 6 ϩ 3 p 5 Ϫ 2 60. 11 p 4 ϩ 7 Ϫ 20 4 61. 12 p 2 Ϫ 3 p 4 62. 5 Ϫ 10 p 6 ϩ 1 63. 2 Ϫ (3 ϩ 4) p 5

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