Angle Bisectors of Triangles

Transcription

1 6 What You ll Learn You ll learn to identify and use angle bisectors in triangles. ngle isectors of Triangles ecall that the bisector of an angle is a ray that separates the angle into two congruent angles. Why It s Important ngineering ngle bisectors of triangles can be found in bridges. ee xercise 9. bisects. m m n angle bisector of a triangle is a segment that separates an angle of the triangle into two congruent angles. One of the endpoints of an angle bisector is a vertex of the triangle, and the other endpoint is on the side opposite that vertex. is an angle bisector of. m m Just as every triangle has three medians, three altitudes, and three perpendicular bisectors, every triangle has three angle bisectors. egment Type roperty pecial egments in Triangles altitude line segment from the vertex, a line perpendicular to the opposite side perpendicular bisector line line segment bisects the side of a triangle angle bisector ray line segment bisects the angle of a triangle n angle bisector of a triangle has all of the characteristics of any angle bisector. In H, J bisects H.., so m m.. m (m H) or (m ) m H. m (m H) or (m ) m H J H 0 hapter 6 ore bout Triangles

2 xamples In, O bisects. If m, find m. ince O bisects, m m. O ince m, m. In, bisects. If m 70, what is m? m (m ) m (70) m 5 efinition of bisector ubstitution ultiply. In, bisects. If m, find m. m (m ) m () m 86 efinition of bisector ubstitution ultiply. Your Turn In, bisects. a. If m, find m. b. ind m if m 5. c. What is m if m 8? lgebra Link U is an angle bisector. In T, ind m UT. T lgebra eview olving ulti-tep quations, p. 7 m UT m U 5x x 5 5x x x 5 x x 5 x 5 x 5 (x 5) U 5x efinition of bisector ubstitution ubtract x from each side. implify. ivide each side by. implify. o, m UT 5(5) or 5. Lesson 6 ngle isectors of Triangles

3 heck for Understanding ommunicating athematics uided ractice xamples. escribe an angle bisector of a triangle.. raw an acute scalene triangle. Then use a compass and straightedge to construct the angle bisector of one of the angles. angle bisector In, bisects, and bisects. H. If m, what is m? H. ind m if m What is m if m? xample 6. lgebra In XYZ, Z W bisects YZX. If m 5x 9 and m 9, find x. X W Y Z xercises ractice Homework Help or xercises 7,, 5, 7 ee xamples 8, 0,,, 6, 0 9, 8 9, xtra ractice ee page 76. In, bisects, and bisects. 7. If m 55, what is m? 8. ind m if m What is m if m? 0. ind m if m 8.. What is m if m 0? In, bisects, bisects, and bisects.. ind m if m.. If m, what is m 6?. What is m if m 6? 5 6 Y 6. What is m if is a right angle? 7. In XYZ, Y W bisects XYZ. What is m XYZ if m 6? 5. ind m if m. pplications and roblem olving X W Z 8. lgebra In, is an angle bisector. If m x 0 and m x 5, find m. hapter 6 ore bout Triangles

4 9. ngineering One type of bridge, a cable-stayed bridge is shown. otice that the cable stay anchorage is an angle bisector of each triangle formed by the cables called stays and the roadway. a. uppose m 0, what is m? b. uppose m 8, what is m? cable stay anchorage stays Tallmadge ridge, avannah, eorgia ixed eview 0. ritical Thinking What kind of angles are formed when you bisect an obtuse angle of a triangle? xplain.. Tell whether the red segment in is an altitude, a perpendicular bisector, both, or neither. (Lesson 6 ). In,,, and are medians. ind if 6. (Lesson 6 ). lgebra The measures of the angles of a triangle are x, x, and x 7. ind the measure of each angle. (Lesson 5 ) tandardized Test ractice xercise. hort esponse Triangle has sides that measure 6 feet, 6 feet, and 9 feet. lassify the triangle by its sides. (Lesson 5 ) 5. ultiple hoice ultiply r s by r s. (lgebra eview) r 5rs s r rs 6r s r s uiz > Lessons 6 through 6 In,,, and are medians. (Lesson 6 ). ind if 9.. What is if 5?. If, what is the measure of?. Tell whether O is an altitude, a perpendicular bisector, both, or neither. (Lesson 6 ) O 5. lgebra In, is an angle bisector. If m x 7 and m x, find m. (Lesson 6 ) Lesson 6 ngle isectors of Triangles

5 hapter 6 aterials ruler protractor compass Investigation ircumcenter, entroid, Orthocenter, and Incenter Is there a relationship between the perpendicular bisectors of the sides of a triangle, the medians, the altitudes, and the angle bisectors of a triangle? Let s find out! Investigate. Use construction tools to locate some interesting points on a triangle. a. raw a large acute scalene triangle. b. On a separate sheet of paper, copy the following table. escription of oints Label of oints midpoints of the three sides () circumcenter () centroid () intersection points of altitudes with the sides () orthocenter () midpoints of segments from orthocenter to each vertex () incenter () midpoint of segment joining circumcenter and orthocenter () c. onstruct the perpendicular bisector of each side of your triangle. Label the midpoints,, and. ecord these letters in your table. The circumcenter is the point where the perpendicular bisectors meet. Label this point J and record it. To avoid confusion, erase the perpendicular bisectors, but not the circumcenter. d. raw the medians of your triangle. The point where the medians meet is the centroid. Label this point and record it in your table. rase the medians, but not the centroid. hapter 6 ore bout Triangles

6 e. onstruct the altitudes of the triangle. Label the points where the altitudes intersect the sides,, and. ecord these points. The point where the altitudes meet is the orthocenter. Label this point and record it. rase the altitudes, but not the orthocenter. U V W J f. raw three segments, each having the orthocenter as one endpoint and a vertex of your triangle as the other endpoint. ind the midpoint of each segment. Label the midpoints U, V, and W and record these points in your table. rase the segments. g. onstruct the bisector of each angle of the triangle. The point where the angle bisectors meet is the incenter. Label this point X and record it. rase the angle bisectors, but not the incenter.. You should now have points labeled. ollow these steps to construct a special circle, called a nine-point circle. a. Locate the circumcenter and orthocenter. raw a line segment connecting these two points. isect this line segment. Label the midpoint Z and record it in the table. o not erase this segment. b. raw a circle whose center is point Z and whose radius extends to a midpoint of the side of your triangle. How many of your labeled points lie on or very close to this circle? c. xtend the segment drawn in tep a. This line is called the uler (OY-ler) line. How many points are on the uler line? In this extension, you will determine whether a special circle exists for other types of triangles. Use paper and construction tools to investigate these cases.. an obtuse scalene triangle. a right triangle. an equilateral triangle resenting Your onclusions Here are some ideas to help you present your conclusions to the class. ake a booklet of your constructions. or each triangle, include a table in which all of the points are recorded. esearch Leonhard uler. Write a brief report on his contributions to mathematics, including the nine-point circle. Investigation or more information on the nine-point circle, visit: hapter 6 Investigation What a ircle! 5