23.2 Angle Bisectors of Triangles

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1 Name lass Date 23.2 ngle isectors of Triangles Essential uestion: How can you use angle bisectors to find the point that is equidistant from all the sides of a triangle? Explore Investigating Distance from a oint to a ine Use a ruler, a protractor, and a piece of tracing paper to investigate points on the bisector of an angle. esource ocker Use the ruler to draw a large angle on tracing paper. abel it. Fold the paper so that coincides with. Open the paper. The crease is the bisector of. lot a point on the bisector. Use the ruler to draw several different segments from point to. Measure the lengths of the segments. Then measure the angle each segment makes with. What do you notice about the shortest segment you can draw from point to? Draw the shortest segment you can from point to. Measure its length. How does its length compare with the length of the shortest segment you drew from point to? Houghton Mifflin Harcourt ublishing ompany eflect 1. uppose you choose a point on the bisector of Z and you draw the perpendicular segment from to and the perpendicular segment from to Z. What do you think will be true about these segments? 2. Discussion What do you think is the best way to measure the distance from a point to a line? Why? Module esson 2

2 Explain 1 pplying the ngle isector Theorem and Its onverse The distance from a point to a line is the length of the perpendicular segment from the point to the line. ou will prove the following theorems about angle bisectors and the sides of the angle they bisect in Exercises 16 and 17. ngle isector Theorem If a point is on the bisector an of angle, then it is equidistant from the sides of the angle., so =. onverse of the ngle isector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. =, so Example 1 Find each measure. M 12.8 M M is the bisector of, so M = M = m D, given that m = 112 eflect 74 D 74 ince D = D, D, and D, you know that D bisects by the o, m D = 1 m =. 2 Theorem. Houghton Mifflin Harcourt ublishing ompany 3. In the onverse of the ngle isector Theorem, why is it important to say that the point must be in the interior of the angle? Module esson 2

3 our Turn Find each measure m M, given that m M = M 62 Explain 2 onstructing an Inscribed ircle circle is inscribed in a polygon if each side of the polygon is tangent to the circle. In the figure, circle is inscribed in quadrilateral WZ and this circle is called the incircle (inscribed circle) of the quadrilateral. In order to construct the incircle of a triangle, you need to find the center of the circle. This point is called the incenter of the triangle. Example 2 Use a compass and straightedge to construct the inscribed circle of. W Z tep 1 The center of the inscribed circle must be equidistant from and. What is the set of points equidistant from and? onstruct this set of points. Houghton Mifflin Harcourt ublishing ompany eflect tep 2 The center must also be equidistant from and. What is the set of points equidistant from and? onstruct this set of points. tep 3 The center must lie at the intersection of the two sets of points you constructed. abel this point. tep 4 lace the point of your compass at and open the compass until the pencil just touches a side of. Then draw the inscribed circle. 6. uppose you started by constructing the set of points equidistant from and, and then constructed the set of points equidistant from and. Would you have found the same center point? heck by doing this construction. Module esson 2

4 Explain 3 Using roperties of ngle isectors s you have seen, the angle bisectors of a triangle are concurrent. The point of concurrency is the incenter of the triangle. Incenter Theorem The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. Z = = Z Example 3 V and V are angle bisectors of. Find each measure. the distance from V to V is the incenter of. y the Incenter Theorem, V is equidistant from the sides of. The distance from V to is 7.3. o the distance from V to is also W V m V V is the bisector of. m = 2 ( ) = Triangle um Theorem + + m = 180 ubtract from each side. m = V is the bisector of. m V = 1 2 ( ) = eflect 7. In art, is there another distance you can determine? Explain. our Turn and are angle bisectors of. Find each measure. 8. the distance from to 9. m Houghton Mifflin Harcourt ublishing ompany Module esson 2

5 Elaborate 10. and are the circumcenter and incenter of T, but not necessarily in that order. Which point is the circumcenter? Which point is the incenter? Explain how you can tell without constructing any bisectors. T 11. omplete the table by filling in the blanks to make each statement true. ircumcenter Incenter Definition The point of concurrency of the The point of concurrency of the Distance Equidistant from the Equidistant from the ocation (Inside, Outside, On) an be the triangle lways the triangle 12. Essential uestion heck-in How do you know that the intersection of the bisectors of the angles of a triangle is equidistant from the sides of the triangle? Evaluate: Homework and ractice Houghton Mifflin Harcourt ublishing ompany 1. Use a compass and straightedge to investigate points on the bisector of an angle. On a separate piece of paper, draw a large angle. a. onstruct the bisector of. b. hoose a point on the angle bisector you constructed. abel it. onstruct a perpendicular through to each side of. c. Explain how to use a compass to show that is equidistant from the sides of. Find each measure. 2. V V 3. m M, given that m = 63 W 4.9 Online Homework Hints and Help Extra ractice M Module esson 2

6 4. D 5. m HF, given that m GF = 45 D H G 10.2 F onstruct an inscribed circle for each triangle M N F and EF are angle bisectors of DE. Find each measure. 8. the distance from F to D 9. m FED 17 F G D E T and are angle bisectors of T. Find each measure. 10. the distance from to 11. m T 42 T Houghton Mifflin Harcourt ublishing ompany Module esson 2

7 Find each measure V 6y y + 6 D 5m - 3 U 2m + 9 V 14. m 15. m GDF (2x + 1)º (3x - 9)º M (6y + 3) G 2.7 D (7y - 3) H 2.7 F 16. omplete the following proof of the ngle isector Theorem. Given: rove: bisects., = Houghton Mifflin Harcourt ublishing ompany tatements 1. bisects., easons 3. and are right angles. 3. Definition of perpendicular ll right angles are congruent eflexive roperty of ongruence Triangle ongruence Theorem = 8. ongruent segments have the same length. Module esson 2

8 17. omplete the following proof of the onverse of the ngle isector Theorem. Given: V, VZ Z, V = VZ. rove: V bisects Z. V Z tatements 1. V, VZ Z, V = VZ 1. easons 2. V and VZ are right angles V V V ZV V ZV omplete the following proof of the Incenter Theorem. Given:,, and bisect, and, respectively.,, Z rove: = = Z et be the incenter of. ince lies on the bisector of, = by the Theorem. imilarly, also, so = Z. Therefore, = = Z, by the. 19. city plans to build a firefighter s monument in a triangular park between three streets. Draw a sketch on the figure to show where the city should place the monument so that it is the same distance from all three streets. ustify your sketch. Fillmore treet uchanan treet olk treet Z Houghton Mifflin Harcourt ublishing ompany Image redits: ep oig/ lamy Module esson 2

9 20. school plans to place a flagpole on the lawn so that it is equidistant from Mercer treet and Houston treet. They also want the flagpole to be equidistant from a water fountain at W and a bench at. Find the point F where the school should place the flagpole. Mark the point on the figure and explain your answer. Mercer treet W Houston treet 21. is the incenter of. Determine whether each statement is true or false. elect the correct answer for each lettered part. a. oint must lie on the perpendicular bisector of. True False b. oint must lie on the angle bisector of. True False c. If is 23 mm long, then must be 23 mm long. True False d. If the distance from point to is x, then the distance from point to must be x. True False e. The perpendicular segment from point to is longer than the perpendicular segment from point to. True False Houghton Mifflin Harcourt ublishing ompany Module esson 2

10 H.O.T. Focus on Higher Order Thinking 22. What If? In the Explore, you constructed the angle bisector of acute and found that if a point is on the bisector, then it is equidistant from the sides of the angle. Would you get the same results if were a straight angle? Explain. 23. Explain the Error student was asked to draw the incircle for. He constructed angle bisectors as shown. Then he drew a circle through points,, and. Describe the student s error. esson erformance Task Teresa has just purchased a farm with a field shaped like a right triangle. The triangle has the measurements shown in the diagram. Teresa plans to install central pivot irrigation in the field. In this type of irrigation, a circular region of land is irrigated by a long arm of sprinklers the radius of the circle that rotates around a central pivot point like the hands of a clock, dispensing water as it moves. a. Describe how she can find where to locate the pivot. 24 yd 51 yd 45 yd b. Find the area of the irrigation circle. To find the radius, r, of a circle inscribed in a triangle with sides of length a, b, and c, you can use the formula r = k = 1 (a + b + c). 2 k (k - a) (k - b) (k - c) k c. bout how much of the field that not be irrigated?, where Houghton Mifflin Harcourt ublishing ompany Module esson 2