3. What is the simplified form of the Midpoint Formula if one of the endpoints of a segment is (0, 0) and the other is (x, y)?

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1 age 5 of 9 GUI TI Vocabulary heck oncept heck kill heck. What kind of geometric figure is an angle bisector?. How do you indicate congruent segments in a diagram? How do you indicate congruent angles in a diagram? 3. What is the simplified form of the Midpoint ormula if one of the endpoints of a segment is (0, 0) and the other is (x, y)? ind the coordinates of the midpoint of a segment with the given endpoints.. (5, ), (º3, ) 5. (º, º9), (, º5) 6. (6, º), (, 8) ind the coordinates of the other endpoint of a segment with the given endpoint and midpoint M. 7. (3, 0) 8. (5, ) 9. (º, ) M(3, ) M(7, 6) M(º3, º) 0. uppose m JKL is 90. If the ray KM Æ bisects JKL, what are the measures of JKM and LKM? Q Æ is the angle bisector of Q. ind the two angle measures not given in the diagram TI N LITION TUNT HL xtra ractice to help you master skills is on p. 80. ONTUTION Use a ruler to measure and redraw the line segment on a piece of paper. Then use construction tools to construct a segment bisector TUNT HL HOMWOK HL xample : xs. 7 xample : xs xample 3: xs. 37 xample : xs. 37 xample 5: xs. 9 INING TH MIOINT ind the coordinates of the midpoint of a segment with the given endpoints. 7. (0, 0) 8. J(º, 7) 9. (0, 8) 0. (º, º9) (º8, 6) K(3, º3) (º, 5) Q(, 0). (0, º8). (, ) 3. V(º.5, 8). G(º5.5, º6.) T(º6, ) (, º8) W(0.5, º) H(º0.5, 9.) 38 hapter asics of Geometry

2 age 6 of 9 xy UING LG ind the coordinates of the other endpoint of a segment with the given endpoint and midpoint M. 5. (, 6) 6. T(º8, º) 7. W(3, º) M(º, ) M(0, 3) M(, º) 8. Q(º5, 9) 9. (6, 7) 30. (º3.5, º6) M(º8, º) M(0, º7) M(.5,.5) OGNIZING ONGUN Use the marks on the diagram to name the congruent segments and congruent angles Z G W X Y ONTUTION Use a protractor to measure and redraw the angle on a piece of paper. Then use construction tools to find the angle bisector NLYZING NGL ITO Q Æ is the angle bisector of Q. ind the two angle measures not given in the diagram TUNT HL OTW HL Visit our Web site to see instructions for several software applications. INTNT 3. THNOLOGY Use geometry software to draw a triangle. onstruct the angle bisector of one angle. Then find the midpoint of the opposite side of the triangle. hange your triangle and observe what happens. oes the angle bisector always pass through the midpoint of the opposite side? oes it ever pass through the midpoint?.5 egment and ngle isectors 39

3 age 7 of 9 TUNT HL xy UING LG Æ bisects. ind the value of x. HOMWOK HL Visit our Web site for help with x. 9. INTNT (x 5) (5x ) (x 5) (x 35) (0x 5) (6x ) (x 9) (x 7) (5x 8) (3x ) x 0 (3x 85) TIK ZON In xercises 50 and 5, use the information below. or each player, find the coordinate of T, a point on the top of the strike zone. In baseball, the strike zone is the region a baseball needs to pass through in order for an umpire to declare it a strike if it is not hit. The top of the strike zone is a horizontal plane passing through the midpoint between the top of the hitter s shoulders and the top of the uniform pants when the player is in a batting stance. ource: Major League aseball T 63 T I HOKY When an air hockey puck is hit into the sideboards, it bounces off so that and are congruent. ind m, m, m 3, and m hapter asics of Geometry

4 age 8 of ILN The diagram represents an unfolded piece of paper used to make a paper airplane. The segments represent where the paper was folded to make the airplane. Using the diagram, name as many pairs of congruent segments and as many congruent angles as you can. N L M K J I G H 56. Writing xplain, in your own words, how you would divide a line segment into four congruent segments using a compass and straightedge. Then explain how you could do it using the Midpoint ormula. 57. MIOINT OMUL VIIT nother version of the Midpoint ormula, for (x, y ) and (x, y ), is M x + (x º x ), y + (y º y ). Test reparation hallenge edo xercises 7 using this version of the Midpoint ormula. o you get the same answers as before? Use algebra to explain why the formula above is euivalent to the one in the lesson. 58. MULTI-T OLM ketch a triangle with three sides of different lengths. a. Using construction tools, find the midpoints of all three sides and the angle bisectors of all three angles of your triangle. b. etermine whether or not the angle bisectors pass through the midpoints. c. Writing Write a brief paragraph explaining your results. etermine if your results would be different if you used a different kind of triangle. ININIT I football team practices running back and forth on the field in a special way. irst they run from one end of the 00 yd field to the other. Then they turn around and run half the previous distance. Then they turn around again and run half the previous distance, and so on. 59. uppose the athletes continue the running drill with smaller and smaller distances. What is the coordinate of the point that they approach? 60. What is the total distance that the athletes cover? XT HLLNG egment and ngle isectors

5 age 9 of 9 MIX VIW KTHING VIUL TTN ketch the next figure in the pattern. (eview.) ITN OMUL ind the distance between the two points. (eview.3) 63. (3, ), (º5, º) 6. (º6, 9), (º, º7) 65. (8, º8), (, ) 66. G(3, º8), H(0, º) 67. J(º, º5), K(5, º) 68. L(º0, ), M(º, 9) MUING NGL Use a protractor to find the measure of the angle. (eview. for.6) QUIZ elf-test for Lessons. and.5. tate the ngle ddition ostulate for the three angles shown at the right. (Lesson.) In a coordinate plane, plot the points and sketch. lassify the angle. Write the coordinates of a point that lies in the interior of the angle and the coordinates of a point that lies in the exterior of the angle. (Lesson.). (º, 3) 3. (º6, º3). (º, 8) 5. (, 0) (, º3) (0, º5) (º, 0) (, ) (, 6) (8, º5) (, 0) (8, ) 6. In the diagram, KM Æ is the angle bisector of JKL. ind m MKL and m JKL. (Lesson.5) J M L K hapter asics of Geometry