5.1. Perpendiculars and Bisectors. What you should learn

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1 age 1 of erpendiculars and isectors What you should learn GOL 1 Use properties of perpendicular bisectors. GOL 2 Use properties of angle bisectors to identify equal distances, such as the lengths of beams in a roof truss in Example 3. Why you should learn it To solve real-life problems, such as deciding where a hockey goalie should be positioned in Exs REL LIFE GOL 1 USING ROERTIES OF ERENIULR ISETORS In Lesson 1.5, you learned that a segment bisector intersects a segment at its midpoint. segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. The construction below shows how to draw a line that is perpendicular to a given line or segment at a point. You can use this method to construct a perpendicular bisector of a segment, as described below the activity. TIVITY onstruction given segment perpendicular bisector is a fi bisector of. erpendicular Through a oint on a Line Use these steps to construct a line that is perpendicular to a given line m and that passes through a given point on m. m m m 1 lace the compass point at. raw an arc that intersects line m twice. Label the intersections as and. 2 Use a compass setting greater than. raw an arc from. With the same setting, draw an arc from. Label the intersection of the arcs as. 3 Use a straightedge to draw. This line is perpendicular to line m and passes through. TIVITY ONSTRUTION STUENT HEL Look ack For a construction of a perpendicular to a line through a point not on the given line, see p You can measure on your construction to verify that the constructed line is perpendicular to the given line m. In the construction, fi and =, so is the perpendicular bisector of. point is equidistant from two points if its distance from each point is the same. In the construction above, is equidistant from and because was drawn so that =. 264 hapter 5 roperties of Triangles

2 age 2 of 8 Theorem 5.1 below states that any point on the perpendicular bisector in the construction is equidistant from and, the endpoints of the segment. The converse helps you prove that a given point lies on a perpendicular bisector. THEOREMS THEOREM 5.1 erpendicular isector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If is the perpendicular bisector of, then =. = THEOREM 5.2 onverse of the erpendicular isector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If =, then lies on the THEOREM perpendicular bisector of. is on. roof lan for roof of Theorem 5.1 Refer to the diagram for Theorem 5.1 above. Suppose that you are given that is the perpendicular bisector of. Show that right triangles and are congruent using the SS ongruence ostulate. Then show that. Exercise 28 asks you to write a two-column proof of Theorem 5.1 using this plan for proof. Exercise 29 asks you to write a proof of Theorem 5.2. EXMLE 1 Using erpendicular isectors Logical Reasoning In the diagram shown, MN is the perpendicular bisector of ST. a. What segment lengths in the diagram are equal? b. Explain why Q is on MN. SOLUTION a. MN bisects ST, so NS = NT. ecause M is on the perpendicular bisector of ST, MS = MT (by Theorem 5.1). The diagram shows that QS = QT = 12. M N q b. QS = QT, so Q is equidistant from S and T. y Theorem 5.2, Q is on the perpendicular bisector of ST, which is MN. T S erpendiculars and isectors 265

3 age 3 of 8 GOL 2 USING ROERTIES OF NGLE ISETORS The distance from a point to a line is defined as the length of the perpendicular segment from the point to the line. For instance, in the diagram shown, the distance between the point Q and the line m is Q. When a point is the same distance from one line as it is from another line, then the point is equidistant from the two lines (or rays or segments). The theorems below show that a point in the interior of an angle is equidistant from the sides of the angle if and only if the point is on the bisector of the angle. q m THEOREMS THEOREM 5.3 ngle isector Theorem If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If m = m, then =. = THEOREM 5.4 THEOREM onverse of the ngle isector Theorem If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. If =, then m = m. paragraph proof of Theorem 5.3 is given in Example 2. Exercise 32 asks you to write a proof of Theorem 5.4. m = m EXMLE 2 roof of Theorem 5.3 roof GIVEN is on the bisector of. fi, fi ROVE = lan for roof rove that. Then conclude that, so =. SOLUTION aragraph roof y the definition of an angle bisector,. ecause and are right angles,. y the Reflexive roperty of ongruence,. Then by the S ongruence Theorem. ecause corresponding parts of congruent triangles are congruent,. y the definition of congruent segments, =. 266 hapter 5 roperties of Triangles

4 age 4 of 8 FOUS ON REERS EXMLE 3 Using ngle isectors ROOF TRUSSES Some roofs are built with wooden trusses that are assembled in a factory and shipped to the building site. In the diagram of the roof truss shown below, you are given that bisects and that and are right angles. What can you say about and? ENGINEERING TEHNIIN In manufacturing, engineering technicians prepare specifications for products such as roof trusses, and devise and run tests for quality control. REER LINK REL INTERNET LIFE SOLUTION ecause and meet and at right angles, they are perpendicular segments to the sides of. This implies that their lengths represent the distances from the point to and. ecause point is on the bisector of, it is equidistant from the sides of the angle. So, =, and you can conclude that. GUIE RTIE Vocabulary heck oncept heck Skill heck 1. If is on the? of, then is equidistant from and. 2. oint G is in the interior of HJK and is equidistant from the sides of the angle, JH and. JK What can you conclude about G? Use a sketch to support your answer. In the diagram, is the perpendicular bisector of. 3. What is the relationship between and? 4. What is the relationship between and? 5. What is the relationship between and? Explain your answer. In the diagram, M is the bisector of LN. 6. What is the relationship between LM and NM? 7. How is the distance between point M and L related to the distance between point M and N? L M N 5.1 erpendiculars and isectors 267

5 age 5 of 8 RTIE N LITIONS STUENT HEL Extra ractice to help you master skills is on p LOGIL RESONING Tell whether the information in the diagram allows you to conclude that is on the perpendicular bisector of. Explain your reasoning LOGIL RESONING In Exercises 11 13, tell whether the information in the diagram allows you to conclude that is on the bisector of. Explain ONSTRUTION raw with a length of 8 centimeters. onstruct a perpendicular bisector and draw a point on the bisector so that the distance between and is 3 centimeters. Measure and. 15. ONSTRUTION raw a large with a measure of 60. onstruct the angle bisector and draw a point on the bisector so that = 3 inches. raw perpendicular segments from to the sides of. Measure these segments to find the distance between and the sides of. USING ERENIULR ISETORS Use the diagram shown. 16. In the diagram, SV fi RT and VR VT. Find VT. R 14 U 17. In the diagram, SV fi RT and VR VT. Find SR. 18. In the diagram, SV is the perpendicular bisector of RT. ecause UR = UT = 14, what can you conclude about point U? S 8 17 V T 14 STUENT HEL HOMEWORK HEL Example 1: Exs. 8 10, 14, 16 18, Example 2: Exs , 15, 19, 20, Example 3: Exs. 31, USING NGLE ISETORS Use the diagram shown. 19. In the diagram, JN bisects HJK, N fi, J NQ fi, JQ and N = 2. Find NQ. 20. In the diagram, JN bisects HJK, MH fi JH, J MK fi JK, and MH = MK = 6. What can you conclude about point M? 2 N q K H 6 6 M 268 hapter 5 roperties of Triangles

6 age 6 of 8 USING ISETOR THEOREMS In Exercises 21 26, match the angle measure or segment length described with its correct value W U E. 50 F X 21. SW 22. m XTV m VWX 24. VU S V T 25. WX 26. m WVX STUENT HEL Look ack For help with proving that constructions are valid, see p THE WRIGHT ROTHERS In Kitty Hawk, North arolina, on ecember 17, 1903, Orville and Wilbur Wright became the first people to successfully fly an engine-driven, heavier-thanair machine. REL LIFE FOUS ON EOLE 27. ROVING ONSTRUTION Write a proof to verify that fi in the construction on page ROVING THEOREM 5.1 Write a proof of Theorem 5.1, the erpendicular isector Theorem. You may want to use the plan for proof given on page 265. GIVEN is the perpendicular bisector of. ROVE is equidistant from and. 29. ROVING THEOREM 5.2 Use the diagram shown to write a two-column proof of Theorem 5.2, the onverse of the erpendicular isector Theorem. GIVEN is equidistant from and. ROVE is on the perpendicular bisector of. lan for roof Use the erpendicular ostulate to draw fi. Show that by the HL ongruence Theorem. Then, so =. 30. ROOF Use the diagram shown. GIVEN GJ is the perpendicular bisector of HK. ROVE GHM GKM 31. ERLY IRRFT On many of the earliest airplanes, wires connected vertical posts to the edges of the wings, which were wooden frames covered with cloth. Suppose the lengths of the wires from the top of a post to the edges of the frame are the same and the distances from the bottom of the post to the ends of the two wires are the same. What does that tell you about the post and the section of frame between the ends of the wires? G M H J K 5.1 erpendiculars and isectors 269

7 age 7 of EVELOING ROOF Use the diagram to complete the proof of Theorem 5.4, the onverse of the ngle isector Theorem. GIVEN is in the interior of and is equidistant from and ROVE lies on the angle bisector of. Statements 1. is in the interior of. 1.? 2. is? from and 2. Given 3.? =? 3. efinition of equidistant 4. fi,?? fi 4. efinition of distance from a point to a line 5.? 5. If 2 lines are fi, then they form 4 rt.. 6.? 6. efinition of right triangle 7. 7.? 8.? 8. HL ongruence Thm ? 10. bisects and point 10.? is on the bisector of. IE HOKEY In Exercises 33 35, use the following information. In the diagram, the goalie is at point G and the puck is at point. The goalie s job is to prevent the puck from entering the goal. 33. When the puck is at the other end of the rink, the goalie is likely to be standing on line l. How is l related to? 34. s an opposing player with the puck skates toward the goal, the goalie is likely to move from line l to other places on the ice. What should be the relationship between G and? 35. How does m change as the puck gets closer to the goal? oes this change make it easier or more difficult for the goalie to defend the goal? Explain. Reasons l goal goal line G 36. TEHNOLOGY Use geometry software to construct. Find the midpoint. raw the perpendicular bisector of through. onstruct a point along the perpendicular bisector and measure and Move along the perpendicular bisector. What theorem does this construction demonstrate? hapter 5 roperties of Triangles

8 age 8 of 8 Test reparation hallenge EXTR HLLENGE MULTI-STE ROLEM Use the map shown and the following information. town planner is trying to decide whether a new household X should be covered by fire station,, or. a. Trace the map and draw the segments,, and. b. onstruct the perpendicular bisectors of,, and. o the perpendicular bisectors meet at a point? c. The perpendicular bisectors divide the town into regions. Shade the region closest to fire station red. Shade the region closest to fire station blue. Shade the region closest to fire station gray. d. Writing In an emergency at household X, which fire station should respond? Explain your choice. xy USING LGER Use the graph at the right. 38. Use slopes to show that WS fi YX and that WT fi YZ. 39. Find WS and WT. 40. Explain how you know that YW bisects XYZ. 1 y S(3, 5) Y (2, 2) 1 X T(5, 1) X(4, 8) W(6, 4) Z(8, 0) x MIXE REVIEW IRLES Find the missing measurement for the circle shown. Use 3.14 as an approximation for π. (Review 1.7 for 5.2) 41. radius 42. circumference 43. area 12 cm LULTING SLOE Find the slope of the line that passes through the given points. (Review 3.6) 44. (º1, 5), (º2, 10) 45. (4, º3), (º6, 5) 46. E(4, 5), F(9, 5) 47. G(0, 8), H(º7, 0) 48. J(3, 11), K(º10, 12) 49. L(º3, º8), M(8, º8) xy USING LGER Find the value of x. (Review 4.1) (2x 6) x x 31 x 70 (10x 22) 5.1 erpendiculars and isectors 271