6.1. Perpendicular and Angle Bisectors


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1 6.1 T TI KOW KI TI TOO To be proficient in math, you need to visualize the results of varying assumptions, explore consequences, and compare predictions with data. erpendicular and ngle isectors ssential uestion What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle? oints on a erpendicular isector Work with a partner. Use dynamic geometry software. a. raw any segment and label it. onstruct the perpendicular bisector of. b. abel a point that is on the perpendicular bisector of but is not on c. raw and and find their lengths. Then move point to other locations on the perpendicular bisector and note the lengths of and. d. epeat parts (a) (c) with other segments. escribe any relationship(s) you notice. oints on an ngle isector Work with a partner. Use dynamic geometry software. a. raw two rays and to form. onstruct the bisector of. b. abel a point on the bisector of. c. onstruct and find the lengths of the perpendicular segments from to the sides of. ove point along the angle bisector and note how the lengths change. d. epeat parts (a) (c) with other angles. escribe any relationship(s) you notice ample oints (1, 3) (2, 1) (2.95, 2.73) egments = 2.24 =? =? ine x + 2y = ample oints (1, 1) (2, 2) (2, 1) (4, 2.24) ays = x + y = 0 = y = 1 ine 0.38x y = 0.54 ommunicate Your nswer 3. What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle? 4. In xploration 2, what is the distance from point to from to is 5 units? Justify your answer. when the distance ection 6.1 erpendicular and ngle isectors 305
2 6.1 esson What You Will earn ore Vocabulary equidistant, p. 306 revious perpendicular bisector angle bisector Use perpendicular bisectors to find measures. Use angle bisectors to find measures and distance relationships. Write equations for perpendicular bisectors. Using erpendicular isectors In ection 3.4, you learned that a perpendicular bisector of a line segment is the line that is perpendicular to the segment at its midpoint. point is equidistant from two figures when the point is the same distance from each figure. TUY TI perpendicular bisector can be a segment, a ray, a line, or a plane. Theorems Theorem 6.1 erpendicular isector Theorem In a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If is the bisector of, then =. is a bisector of. roof p. 306; x. 21, p. 352 Theorem 6.2 onverse of the erpendicular isector Theorem In a plane, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment. If =, then point lies on the bisector of. roof x. 32, p. 312 erpendicular isector Theorem iven is the perpendicular bisector of. rove = 306 hapter 6 elationships Within Triangles aragraph roof ecause is the perpendicular bisector of, is perpendicular to and point is the midpoint of. y the definition of midpoint, =, and by the definition of perpendicular lines, m = m = 90. Then by the definition of segment congruence,, and by the definition of angle congruence,. y the eflexive roperty of ongruence (Theorem 2.1),. o, by the ongruence Theorem (Theorem 5.5), and because corresponding parts of congruent triangles are congruent. o, = by the definition of segment congruence.
3 Using the erpendicular isector Theorems ind each measure. a. rom the figure, is the perpendicular bisector of. y the erpendicular isector Theorem, =. b. o, = = 6.8. ecause = and, is the perpendicular bisector of by the onverse of the erpendicular isector Theorem. y the definition of segment bisector, = 2. o, = 2(9.5) = c. rom the figure, is the perpendicular bisector of. 3x + 14 = 5x = 3x + 14 erpendicular isector Theorem ubstitute. x = 7 olve for x. o, = 5x = 5(7) = 35. 5x olving a ealife roblem K Is there enough information in the diagram to conclude that point lies on the perpendicular bisector of K? OUTIO It is given that K. o, is a segment bisector of K. You do not know whether is perpendicular to K because it is not indicated in the diagram. o, you cannot conclude that point lies on the perpendicular bisector of K. onitoring rogress Use the diagram and the given information to find the indicated measure. 1. Z is the perpendicular bisector of WY, and YZ = ind WZ. 2. Z is the perpendicular bisector of WY, WZ = 4n 13, and YZ = n ind YZ. 3. ind W when WZ = 20.5, WY = 14.8, and YZ = elp in nglish and panish at igideasath.com Z W Y ection 6.1 erpendicular and ngle isectors 307
4 Using ngle isectors In ection 1.5, you learned that an angle bisector is a ray that divides an angle into two congruent adjacent angles. You also know that the distance from a point to a line is the length of the perpendicular segment from the point to the line. o, in the figure, is the bisector of, and the distance from point to is, where. Theorems Theorem 6.3 ngle isector Theorem If a point lies on the bisector of an angle, then it is equidistant from the two sides of the angle. If bisects and and, then =. roof x. 33(a), p. 312 Theorem 6.4 onverse of the ngle isector Theorem If a point is in the interior of an angle and is equidistant from the two sides of the angle, then it lies on the bisector of the angle. If and and =, then bisects. roof x. 33(b), p. 312 Using the ngle isector Theorems ind each measure. a. m J ecause J and J and J = J = 7, J bisects by the onverse of the ngle isector Theorem. o, m J = m J = 42. J b. = 5x = 6x 5 ngle isector Theorem ubstitute. 5x 6x 5 5 = x olve for x. o, = 6x 5 = 6(5) 5 = 25. onitoring rogress 308 hapter 6 elationships Within Triangles elp in nglish and panish at igideasath.com Use the diagram and the given information to find the indicated measure. 4. bisects, and = 6.9. ind. 5. bisects, = 3z + 7, and = 2z ind. 6. ind m when = 3.2, = 3.2, and m = 39.
5 olving a ealife roblem soccer goalie s position relative to the ball and goalposts forms congruent angles, as shown. Will the goalie have to move farther to block a shot toward the right goalpost or the left goalpost? OUTIO The congruent angles tell you that the goalie is on the bisector of. y the ngle isector Theorem, the goalie is equidistant from and. o, the goalie must move the same distance to block either shot. Writing quations for erpendicular isectors Writing an quation for a isector y y = 3x 1 (1, 2) 2 4 x Write an equation of the perpendicular bisector of the segment with endpoints ( 2, 3) and (4, 1). OUTIO tep 1 raph. y definition, the perpendicular bisector of is perpendicular to at its midpoint. tep 2 ind the midpoint of. ( , 2 2 ) ( = 2 2, 4 = (1, 2) 2) tep 3 ind the slope of the perpendicular bisector. slope of 1 3 = 4 ( 2) = 2 6 = 1 3 ecause the slopes of perpendicular lines are negative reciprocals, the slope of the perpendicular bisector is 3. tep 4 Write an equation. The bisector of has slope 3 and passes through (1, 2). y = mx + b Use slopeintercept form. 2 = 3(1) + b ubstitute for m, x, and y. 1 = b olve for b. o, an equation of the perpendicular bisector of is y = 3x 1. onitoring rogress elp in nglish and panish at igideasath.com 7. o you have enough information to conclude that bisects? xplain. 8. Write an equation of the perpendicular bisector of the segment with endpoints ( 1, 5) and (3, 1). ection 6.1 erpendicular and ngle isectors 309
6 6.1 xercises ynamic olutions available at igideasath.com Vocabulary and ore oncept heck 1. OT T T oint is in the interior of. If and are congruent, then is the of. 2. IT WO, UTIO Which is different? ind both answers. Is point the same distance from both and Z? Is point equidistant from and Z? Is point collinear with and Z? Is point on the perpendicular bisector of Z? Z onitoring rogress and odeling with athematics In xercises 3 6, find the indicated measure. xplain your reasoning. (ee xample 1.) K UW J 5x 4x V 9x + 1 U 4.7 T W 7x In xercises 11 14, find the indicated measure. xplain your reasoning. (ee xample 3.) 11. m In xercises 7 10, tell whether the information in the diagram allows you to conclude that point lies on the perpendicular bisector of. xplain your reasoning. (ee xample 2.) K 13. m KJ 14. K 7x J (3x + 16) x x hapter 6 elationships Within Triangles
7 In xercises 15 and 16, tell whether the information in the diagram allows you to conclude that bisects. xplain your reasoning. (ee xample 4.) OI WIT TTI The diagram shows the position of the goalie and the puck during a hockey game. The goalie is at point, and the puck is at point. In xercises 17 and 18, tell whether the information in the diagram allows you to conclude that =. xplain your reasoning In xercises 19 22, write an equation of the perpendicular bisector of the segment with the given endpoints. (ee xample 5.) 19. (1, 5), (7, 1) 20. ( 2, 0), (6, 12) 21. U( 3, 4), V(9, 8) 22. Y(10, 7), Z( 4, 1) O YI In xercises 23 and 24, describe and correct the error in the student s reasoning ecause =, will pass through point. y the ngle isector Theorem (Theorem 6.3), x = OI TTI In the photo, the road is perpendicular to the support beam and. Which theorem allows you to conclude that? 5 x goal goal line a. What should be the relationship between and to give the goalie equal distances to travel on each side of? b. ow 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? xplain your reasoning. 27. OTUTIO Use a compass and straightedge to construct a copy of Y. onstruct a perpendicular bisector and plot a point Z on the bisector so that the distance between point Z and Y is 3 centimeters. easure Z and YZ. Which theorem does this construction demonstrate? 28. WITI xplain how the onverse of the erpendicular isector Theorem (Theorem 6.2) is related to the construction of a perpendicular bisector. 29. OI What is the value of x in the diagram? not enough information 30. OI Which point lies on the perpendicular bisector of the segment with endpoints (7, 5) and ( 1, 5)? (2, 0) (3, 9) (4, 1) (1, 3) 31. KI UT Your friend says it is impossible for an angle bisector of a triangle to be the same line as the perpendicular bisector of the opposite side. Is your friend correct? xplain your reasoning. Y (3x 9) ection 6.1 erpendicular and ngle isectors 311
8 32. OVI TO rove the onverse of the erpendicular isector Theorem (Thm. 6.2). (int: onstruct a line through point perpendicular to at point.) iven = rove oint lies on the perpendicular bisector of. 33. OVI TO Use a congruence theorem to prove each theorem. a. ngle isector Theorem (Thm. 6.3) b. onverse of the ngle isector Theorem (Thm. 6.4) 34. OW O YOU IT? The figure shows a map of a city. The city is arranged so each block north to south is the same length and each block east to west is the same length. 1st t. ercy ospital Wilson chool 2nd t. useum 3rd t. ark t. ain t. cademy chool Oak t. ine t. oosevelt chool aple t. 4th t. Trinity ospital 5th t. W ine treet chool 6th t. 35. TTI OTIO Write an equation whose graph consists of all the points in the given quadrants that are equidistant from the x and yaxes. a. I and III b. II and IV c. I and II 36. TOUT OVOKI The postulates and theorems in this book represent uclidean geometry. In spherical geometry, all points are on the surface of a sphere. line is a circle on the sphere whose diameter is equal to the diameter of the sphere. In spherical geometry, is it possible for two lines to be perpendicular but not bisect each other? xplain your reasoning. 37. OO Use the information in the diagram to prove that if and only if points,, and are collinear. 38. OO rove the statements in parts (a) (c). Y V W a. Which school is approximately equidistant from both hospitals? xplain your reasoning. b. Is the museum approximately equidistant from Wilson chool and oosevelt chool? xplain your reasoning. aintaining athematical roficiency lassify the triangle by its sides. (ection 5.1) Z iven lane is a perpendicular bisector of Z at point Y. rove a. W ZW b. V ZV c. VW VZW eviewing what you learned in previous grades and lessons lassify the triangle by its angles. (ection 5.1) hapter 6 elationships Within Triangles
Relationships Within Triangles
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