5-2 erpendicular and ngle isectors ommon ore tate tandards G-O..9 rove theorems about lines and angles... points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. lso G-..5 1, 3, 4, 5, 8 Objective o use properties of perpendicular bisectors and angle bisectors onfused? ry drawing a diagram to straighten yourself out. HEIL IE You hang a bulletin board over your desk using string. he bulletin board is crooked. When you straighten the bulletin board, what type of triangle does the string form with the top of the board? How do you know? Visualize the vertical line along the wall that passes through the nail. What relationships exist between this line and the top edge of the straightened bulletin board? Explain. In the olve It, you thought about the relationships that must exist in order for a bulletin board to hang straight. You will explore these relationships in this lesson. Lesson L VocabularyV equidistant distance from a point to a line Essential Understanding here is a special relationship between the points on the perpendicular bisector of a segment and the endpoints of the segment. In the diagram below on the left, < > is the perpendicular bisector of. < > is perpendicular to at its midpoint. In the diagram on the right, and are drawn to complete and. You should recognize from your work in hapter 4 that. o you can conclude that, or that =. point is equidistant from two objects if it is the same distance from the objects. o point is equidistant from points and. his suggests a proof of heorem 5-2, the erpendicular isector heorem. Its converse is also true and is stated as heorem 5-3. 292 hapter 5 elationships Within riangles
heorem 5-2 erpendicular isector heorem heorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If... hen... < > # and = = You will prove heorem 5-2 in Exercise 32. heorem 5-3 onverse of the erpendicular isector heorem heorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If... = hen... < > # and = You will prove heorem 5-3 in Exercise 33. roblem 1 Using the erpendicular isector heorem How do you know is the perpendicular bisector of? he markings in the diagram show that is perpendicular to at the midpoint of. lgebra What is the length of? is the perpendicular bisector of, so is equidistant from and. = erpendicular isector heorem 4x = 6x - 10 ubstitute 4x for and 6x - 10 for. - 2x = - 10 ubtract 6x from each side. 4x 6x 10 x = 5 ivide each side by - 2. Now find. = 4x = 4(5) = 20 ubstitute 5 for x. Got It? 1. What is the length of? 3n 1 5n 7 Lesson 5-2 erpendicular and ngle isectors 293
roblem 2 How do you find points that are equidistant from two given points? y the onverse of the erpendicular isector heorem, points equidistant from two given points are on the perpendicular bisector of the segment that joins the two points. Using a erpendicular isector park director wants to build a -shirt stand equidistant from the ollin oaster and the paceship hoot. What are the possible locations of the stand? Explain. addle boats paceship hoot ollin oaster erry-go-round o be equidistant from the two rides, the stand should be on the perpendicular bisector of the segment connecting the rides. Find the midpoint of and draw line / through perpendicular to. he possible locations of the stand are all the points on line /. Got It? 2. a. uppose the director wants the -shirt stand to be equidistant from the paddle boats and the paceship hoot. What are the possible locations? b. easoning an you place the -shirt stand so that it is equidistant from hsm11gmse_0502_t13165 the paddle boats, the paceship hoot, and the ollin oaster? Explain. Essential Understanding here is a special relationship between the points on the bisector of an angle and the sides of the angle. he distance from a point to a line is the length of the perpendicular segment from the point to the line. his distance is also the length of the shortest segment from the point to the line. You will prove this in Lesson 5-6. In the figure at the right, the distances from to / and from to / are represented by the red segments. > 0 1 2 3 4 5 6 In the diagram, is the bisector of. If you measure the lengths of the perpendicular segments from to the two sides of hsm11gmse_0502_t06088 the angle, you will find that the lengths are equal. oint is equidistant from the sides of the angle. 294 hapter 5 elationships Within riangles hsm11gmse_0502_t06089
heorem 5-4 ngle isector heorem heorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. If... > bisects, # >, and # > hen... = You will prove heorem 5-4 in Exercise 34. heorem 5-5 onverse of the ngle isector heorem heorem If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector. If... # >, # >, and = hen... > bisects You will prove heorem 5-5 in Exercise 35. roblem 3 lgebra What is the length of? Using the ngle isector heorem 2x 25 N > bisects LN. # NL > and # N >. he length of Use the ngle isector heorem to write an equation you can solve for x. L 7x = ngle isector heorem 7x = 2x + 25 ubstitute. N How can you use the expression given for to check your answer? ubstitute 5 for x in the expression 2x + 25 and verify that the result is 35. 5x = 25 ubtract 2x from each side. x = 5 ivide each side by 5. Now find. = 7x = 7(5) = 35 ubstitute 5 for x. Got It? 3. What is the length of F? 6x 3 F 4x 9 Lesson 5-2 erpendicular and ngle isectors 295
Lesson heck o you know HOW? Use the figure at the right for Exercises 1 3. 1. What is the relationship between and? 2. What is the length of? 3. What is the length of? 15 E 18 o you UNEN? HEIL IE 4. Vocabulary raw a line and a point not on the line. raw the segment that represents the distance from the point to the line. 5. Writing oint is in the interior of LOX. escribe how you can determine whether is on the bisector of LOX without drawing the angle bisector. ractice and roblem-olving Exercises HEIL IE ractice Use the figure at the right for Exercises 6 8. ee roblem 1. 6. What is the relationship between and JK? 9x 18 3x 7. What is value of x? 8. Find J. J K eading aps For Exercises 9 and 10, use the map of a part of anhattan. 9. Which school is equidistant from the subway stations at Union quare and 14th treet? How do you know? 10. Is t. Vincent s Hospital equidistant from Village Kids Nursery chool and Legacy chool? How do you know? 11. Writing On a piece of paper, mark a point H for home and a point for school. escribe how to find the set of points equidistant from H and. 14th t 8th ve NY useum chool t. Vincent s Hospital 7th ve Village Kids Nursery chool ee roblem 2. = ubway tation 6th ve roadway oleman chool Xavier chool Legacy chool Union quare Use the figure at the right for Exercises 12 15. ee roblem 3. 12. ccording to the diagram, how far is L from HK >? From HF >? 13. How is HL > related to KHF? Explain. 14. Find the value of y. 15. Find m KHL and m FHL. K 27 6y H L F (4y 18) 296 hapter 5 elationships Within riangles
16. lgebra Find x, JK, and J. 17. lgebra Find y,, and U. x 5 K 5y J L 3y 6 2x 7 V U pply lgebra Use the figure at the right for Exercises 18 22. 18. Find the value of x. 19. Find W. 20. Find WZ. 2x Y 21. What kind of triangle is WZ? Explain. 22. If is on the perpendicular bisector of Z, then is? from and Z, or? =?. W 3x 5 Z 23. hink bout a lan In the diagram at the right, the soccer goalie will prepare for a shot from the player at point by moving out to a point on XY. o have the best chance of stopping the ball, should the goalie stand at the point on XY that lies on the perpendicular bisector of GL or at the point on XY that lies on the bisector of GL? Explain your reasoning. How can you draw a diagram to help? Would the goalie want to be the same distance from G and L or from G and L? X G Y L 24. a. onstructions raw E. onstruct the angle bisector of the angle. b. easoning Use the converse of the angle bisector theorem to justify your construction. 25. a. onstructions raw. onstruct the perpendicular bisector of to construct. b. easoning Use the perpendicular bisector theorem to justify that your construction is an isosceles triangle. 26. Write heorems 5-2 and 5-3 as a single biconditional statement. 27. Write heorems 5-4 and 5-5 as a single biconditional statement. Lesson 5-2 erpendicular and ngle isectors 297
28. Error nalysis o prove that is isosceles, a student began by stating that since is on the segment perpendicular to, is equidistant from the endpoints of. What is the error in the student s reasoning? Writing etermine whether must be on the bisector of jx. Explain. 29. X 30. 31. 8 9 X X 32. rove the erpendicular roof isector heorem. Given: < > #, < > bisects rove: = 33. rove the onverse of the roof erpendicular isector heorem. Given: = with # at. rove: is on the perpendicular bisector of. 34. rove the ngle roof isector heorem. Given: > bisects, # >, # > rove: = 35. rove the onverse of the roof ngle isector heorem. Given: # >, # >, = rove: > bisects. 36. oordinate Geometry Use points (6, 8), O(0, 0), and (10, 0). a. Write equations of lines /and m such that / # < O > at and m # < O > at. b. Find the intersection of lines / and m. c. how that =. d. Explain why is on the bisector of O. 298 hapter 5 elationships Within riangles
hallenge 37.,, and are three noncollinear points. escribe and sketch a line in plane such that points,, and are equidistant from the line. Justify your response. E 38. easoning is the intersection of the perpendicular bisectors of two sides of. Line / is perpendicular to plane at. Explain why a point E on / is equidistant from,, and. (Hint: ee page 48, Exercise 33. Explain why E E E.) tandardized est rep / 39. For (1, 3) and (1, 9), which point lies on the perpendicular bisector of? (3, 3) (1, 5) (6, 6) (3, 12) 40. What is the converse of the following conditional statement? If a triangle is isosceles, then it has two congruent angles. If a triangle is isosceles, then it has two congruent sides. If a triangle has congruent sides, then it is equilateral. If a triangle has two congruent angles, then it is isosceles. If a triangle is not isosceles, then it does not have two congruent angles. 41. Which figure represents the statement bisects? hort esponse 42. he line y = 7 is the perpendicular bisector of the segment with endpoints (2, 10) and (2, k). What is the value of k? Explain your reasoning. ixed eview 43. Find the value of x in the figure at the right. 44. 1 and 2 are complementary and 1 and 3 are supplementary. If m 2 = 30, what is m 3? ee Lesson 5-1. 2x 10 3x 7 ee Lesson 1-5. Get eady! o prepare for Lesson 5-3, do Exercises 45 47. 45. What is the slope of a line that is perpendicular to the line y = -3x + 4? 46. Line / is a horizontal line. What is the slope of a line perpendicular to /? 47. escribe the line x = 5. ee Lesson 3-8. Lesson 5-2 erpendicular and ngle isectors 299