12.1 Triangle Proportionality Theorem

Transcription

1 Name lass Date 12.1 Triangle Proportionality Theorem ssential Question: When a line parallel to one side of a triangle intersects the other two sides, how does it divide those sides? Resource Locker xplore onstructing Similar Triangles In the following activity you will see one way to construct a triangle similar to a given triangle. Do your work for Steps in the space provided. Draw a triangle. Label it as shown. heck students' constructions. Select a point on. Label it. onstruct an angle with vertex that is congruent to. Label the point where the side of the angle you constructed intersects as. Houghton Mifflin Harcourt Publishing ompany D Why are and parallel? oplanar lines and are cut by transversal so that. y the onverse of the orresponding ngles Theorem,. Use a ruler to measure,,, and. Then compare the ratios and. Measurements will vary, but, if the constructions are accurate, the ratios should be approximately equal. Module Lesson 1 DO NOT DIT--hanges must be made through "ile info" orrectionkey=nl-;-

2 1. Discussion How can you show that? xplain. You can show that there are three pairs of congruent angles. (Reflexive Property of quality) and (by construction). lso, because,. Then you can use the Similarity riterion because you can show that there are two pairs of congruent angles. 2. What do you know about the ratios and? xplain. ecause and corresponding sides of similar triangles are. Make a onjecture Use your answer to Step to make a conjecture about the line segments produced when a line parallel to one side of a triangle intersects the other two sides. The parallel line divides the other two sides so the lengths of the segments are proportional. xplain 1 proportional, = Proving the Triangle Proportionality Theorem s you saw in the xplore, when a line parallel to one side of a triangle intersects the other two sides of the triangle, the lengths of the segments are proportional. Triangle Proportionality Theorem Theorem Hypothesis onclusion If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. xample 1 Given: Prove: = Prove the Triangle Proportionality Theorem Step 1 Show that. ecause, you can conclude that 1 2 and 4 by the orresponding ngles So, by the Similarity riterion. = Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

3 Step 2 Use the fact that corresponding sides of similar triangles are proportional to prove that =. = orresponding sides are proportional. + + = Segment ddition Postulate 1 + = 1 + Use the property that a + b c = a c + b c. = Subtract 1 from both sides. = Take the reciprocal of both sides. 4. xplain how you conclude that without using and 4. by the Reflexive Property of ongruence, and 1 2 since they are corresponding angles; by the Similarity riterion. xplain 2 pplying the Triangle Proportionality Theorem xample 2 ind the length of each segment. Houghton Mifflin Harcourt Publishing ompany Y It is given that XY X so X = Y by the Triangle Proportionality Y Substitute 9 for X, 4 for X, and 10 for Y. Then solve for Y. 9 4 = 10 Y Take the reciprocal of both sides. 4 9 = Y 10 Next, multiply both sides by ( 4 10 ) = Y, or = Y 9) = ( Y ind PN. It is given that PQ LM, so NQ NP QM = PL by the L P Triangle Proportionality Substitute 5 for NQ, 2 for QM, and for PL. 5 2 = NP 15 Multiply both sides by : 5 ( ) 2 or = NP 2) = ( NP 4 X Y 10 N 9 5 M Q 2 Module 12 6 Lesson 1

4 Your Turn ind the length of each segment. 5. DG D 24 G D 2 = DG ; 24 = DG ; 2 = DG 40 ; 40 ( 24 2 ) = DG; DG = = 0 RN Q 5 P 8 M 10 R MR MQ RN = QP ; 10 8 RN 5 RN = 5 ; 10 = RN = 50 8 = or 6 4 N 8 ; RN = ( 5 8 ) 10 xplain Proving the onverse of the Triangle Proportionality Theorem The converse of the Triangle Proportionality Theorem is also true. onverse of the Triangle Proportionality Theorem Theorem Hypothesis onclusion If a line divides two sides of a triangle proportionally, then it is parallel to the third side. = xample Prove the onverse of the Triangle Proportionality Theorem Given: = Prove: Step 1 Show that. It is given that =, and taking the reciprocal = of both sides shows that. Now add 1 to both sides by adding to the left side and to the right side. This gives + = +. dding and using the Segment ddition Postulate gives =. Since, by the SS Similarity Step 2 Use corresponding angles of similar triangles to show that. and are corresponding angles. So, onverse of the orresponding ngles by the Houghton Mifflin Harcourt Publishing ompany Module Lesson 1

5 R 7. ritique Reasoning student states that UV must be parallel to ST U V. Do you agree? Why or why not? RU S Yes; because RU = US and RV = VS, US = RV VT = 1. So UV ST by the onverse of the Triangle Proportionality T xplain 4 pplying the onverse of the Triangle Proportionality Theorem You can use the onverse of the Triangle Proportionality Theorem to verify that a line is parallel to a side of a triangle. xample 4 Verify that the line segments are parallel. MN and KL JM MK = = 2 JN N L = 0 15 = 2 Since JM MK = JN N L, MN KL by the onverse of the K 21 M 42 0 N 15 L J Triangle Proportionality D and (Given that = 6 cm, and = 27 cm) D = - D = 6-20 = 16 = - = = 12 D 20 cm 15 cm Houghton Mifflin Harcourt Publishing ompany D D = = = = 16 4 Since D D =, D by the 8. ommunicate Mathematical Ideas In, in the example, what is the value of? xplain how you know. V D D = 9 Your Turn 9. Verify that TU and RS are parallel. R VT 90 5 VU 72 TR = 72 = 4, US = = = 4 T VT VU 90 TR = US, so RS TU U 54 S onverse of the Triangle Proportionality D = 5 ; Possible answer: because D, corresponding angles and D are congruent, as are corresponding angles and D. So, D and D = D =. Module Lesson 1

6 laborate 10. In, XY. Use what you know about similarity and proportionality to identify as many different proportions as possible. Possible answers: X = Y ; X = Y ; X X = Y X Y Y ; X = Y X Y 11. Discussion What theorems, properties, or strategies are common to the proof of the Triangle Proportionality Theorem and the proof of onverse of the Triangle Proportionality Theorem? Possible answers: Segment ddition Postulate; properties of fractions; taking reciprocals of both sides of a proportion, and use of Similarity riteria ( for the Triangle Proportionality Theorem and SS for its converse) 12. ssential Question heck-in Suppose a line parallel to side of intersects sides and at points X and Y, respectively, and X = 1. What X do you know about X and Y? xplain. X and Y are the midpoints of sides and. If X = 1, then Y X Y = 1. Then X = X, so X is the midpoint of. Similarly, Y is the midpoint of.