12.1 Triangle Proportionality Theorem

Transcription

1 ame lass Date 12.1 Triangle roportionality 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 ocker 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. abel it as shown. Select a point on. abel it. onstruct an angle with vertex that is congruent to. abel the point where the side of the angle you constructed intersects as. Houghton ifflin Harcourt ublishing ompany D Why are and parallel? Use a ruler to measure,,, and. Then compare the ratios and. odule esson 1

2 Reflect 1. Discussion How can you show that? xplain. 2. What do you know about the ratios and? xplain.. ake 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. xplain 1 roving the Triangle roportionality 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 roportionality 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: rove the Triangle roportionality Theorem rove: = Step 1 Show that. ecause, you can conclude that 1 2 and 4 by the Theorem. = Houghton ifflin Harcourt ublishing ompany So, by the. odule esson 1

3 Step 2 Use the fact that corresponding sides of similar triangles are proportional to prove that =. = orresponding sides are proportional. + = Segment ddition ostulate 1 + = Use the property that a + b c = Subtract 1 from both sides. = a c + b c. = Take the reciprocal of both sides. Reflect 4. xplain how you conclude that without using and 4. xplain 2 pplying the Triangle roportionality Theorem xample 2 ind the length of each segment. Houghton ifflin Harcourt ublishing ompany Y It is given that XY X so X = Y by the Triangle roportionality Y Theorem. 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 ext, multiply both sides by ( 4 10 ) = Y, or = Y 9) = ( Y ind. It is given that Q, so Q Q = Triangle roportionality Theorem. by the Substitute for Q, for Q, and for. 5 2 = ultiply both sides by : 5 ( ) = 2) = ( 4 X Y Q 2 odule 12 6 esson 1

4 Your Turn ind the length of each segment. 5. DG D 24 G R 8 10 Q 5 R xplain roving the onverse of the Triangle roportionality Theorem The converse of the Triangle roportionality Theorem is also true. onverse of the Triangle roportionality Theorem Theorem Hypothesis onclusion If a line divides two sides of a triangle proportionally, then it is parallel to the third side. = xample rove the onverse of the Triangle roportionality Theorem Given: = rove: Step 1 Show that. It is given that =, and taking the reciprocal of both sides shows that. ow add 1 to both sides by adding This gives. to the left side and to the right side. dding and using the Segment ddition ostulate gives. Since, by the Theorem. Step 2 Use corresponding angles of similar triangles to show that. Houghton ifflin Harcourt ublishing ompany and are corresponding angles. So, by the Theorem. odule esson 1

5 Reflect R 7. ritique Reasoning student states that UV must be parallel to ST. Do you agree? Why or why not? S U V T xplain 4 pplying the onverse of the Triangle roportionality Theorem You can use the onverse of the Triangle roportionality Theorem to verify that a line is parallel to a side of a triangle. xample 4 Verify that the line segments are parallel. and K J K = = 2 J = 0 15 = 2 Since J K = J, K by the onverse of the K J Triangle roportionality Theorem. D and (Given that = 6 cm, and = 27 cm) D = - D = 6-20 = 16 = - = - = D 20 cm 15 cm D D = = = = Since D D =, D by the Theorem. Houghton ifflin Harcourt ublishing ompany Reflect 8. ommunicate athematical Ideas In, in the example, what is the value of? xplain how you know. D Your Turn 9. Verify that 90 TU and RS are parallel. T 72 R V 67.5 U 54 S odule esson 1

6 laborate 10. In, XY. Use what you know about similarity and proportionality to identify as many different proportions as possible. X Y 11. Discussion What theorems, properties, or strategies are common to the proof of the Triangle roportionality Theorem and the proof of onverse of the Triangle roportionality Theorem? 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. valuate: Homework and ractice 1. opy the triangle that you drew for the xplore activity. onstruct a line G parallel to using the same method you used in the xplore activity. Online Homework Hints and Help xtra ractice 2. ZY. Write a paragraph proof to show that X Z = X Y. X Y Z Houghton ifflin Harcourt ublishing ompany odule esson 1

7 ind the length of each segment.. K 4. XZ 5. V G 6 J 4 K 8 H m n Y 0 18 U X Z 0 V V U T Verify that the given segments are parallel. 6. and D 7. and QR D 2.7 Q R 4 2 WX and D W X D 9. Use the onverse of the Triangle roportionality Theorem to identify parallel lines in the figure Houghton ifflin Harcourt ublishing ompany 10. On the map, 1st Street and 2nd Street are parallel. What is the distance from ity Hall to 2nd Street along edar Road? ity Hall 2.1 mi 2.8 mi 2.4 mi spen Rd. ibrary edar Rd st St. 2nd St. odule esson 1

8 11. On the map, 5th venue, 6th venue, and 7th venue are parallel. What is the length of ain Street between 5th venue and 6th venue? 5th ve. 6th ve. 7th ve. ain St. 0.4 km Spring St. 0. km 0.5 km 12. ulti-step The storage unit has horizontal siding that is parallel to the base. G a. ind. 11. ft 10.4 ft b. ind G. c. ind to the nearest tenth of a foot. H J 2.2 ft K D 2.6 ft d. ake a onjecture Write the ratios and HJ JK as decimals to the nearest hundredth and compare them. ake a conjecture about the relationship between parallel lines D,, and and transversals G and GK. 1. corollary to the onverse of the Triangle roportionality Theorem states that if three or more parallel lines intersect two transversals, then they divide the transversals proportionally. omplete the proof of the corollary. Given: arallel lines D, D rove: = X X, X X = D D, = D D X D Statements 1. D, D 1. Given 2. Draw intersecting D at X. 2. Two points. = X X 4. X X = D D 5. = D D. 4. Reasons 5. roperty of quality Houghton ifflin Harcourt ublishing ompany Image redits: Stuart Walker/lamy odule esson 1

9 14. Suppose that = 24. Use the Triangle roportionality Theorem to find. 15. Which of the given measures allow you to conclude that UV ST? Select all that apply.. SR = 12, TR = 9. SR = 16, TR = 20. SR = 5, TR = 28 D. SR = 50, TR = 48 K U V S T R. SR = 25, TR = 20 H.O.T. ocus on Higher Order Thinking 16. lgebra or what value of x is G HJ? G 40 H 4x + 4 Houghton ifflin Harcourt ublishing ompany 45 J 5x ommunicate athematical Ideas John used to write a proof of the entroid Theorem. He began by drawing medians K and, intersecting at Z. ext he drew midsegments and, both parallel to median K. Given: with medians K and, and midsegments and rove: Z is located 2 of the distance from each vertex of to the midpoint of the opposite side. a. omplete each statement to justify the first part of John s proof. y the definition of, K = 1 K. y the definition 2 of, K = K. So, by, K = 1 K, or K 2 K = 2. onsider. K (and therefore ZK ), so Z Z = K by the K Theorem, and Z = 2Z. ecause = Z, Z = 2Z Z = 2, and Z is located 2 of the distance from vertex of to the midpoint of the opposite side. Z K Z K b. xplain how John can complete his proof. odule esson 1

10 18. ersevere in roblem Solving Given with = 5, you want to find. irst, find the value that y must have for the Triangle roportionality Theorem to apply. Then describe more than one way to find, and find. y 9 (5.5, y) (7, 6) (1, 2) (4, 2) (10, 2) x esson erformance Task Shown here is a triangular striped sail, together with some of its dimensions. In the diagram, segments J, I, and DH are all parallel to segment G. ind each of the following: 1. J 2. D. HG 2.5 ft 2.25 ft J 1.8 ft I 1.2 ft H 4. G D 5. the perimeter of 6. the area of 7. the number of sails you could make for $10,000 if the sail material costs $0 per square yard.5 ft 6 ft 6.5 ft G Houghton ifflin Harcourt ublishing ompany odule esson 1