7-1. Ratios and Proportions. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

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1 7-1 Ratios and Proportions Vocabulary Review 1. Write a ratio to compare 9 red marbles to 16 blue marbles in three ways. 9 to : 16 In simplest form, write the ratio of vowels to consonants in each word below. 2. comparison 3. geometry 4. ratio to : 5. ross out the ratio that is NOT equivalent to 12 to : 2 9 to : 32 Vocabulary uilder proportion (noun) pruh PWR shun Other Word Form: proportional (adjective) Definition: proportion is an equation stating that two ratios are equal. Eamples: and are proportions. Use Your Vocabulary 6. Write 3 or 6 to make each proportion true Underline the correct word to complete each sentence. 7. Distance on a map is proportion / proportional to the actual distance. 8. The number of ounces in 3 lb is in proportion / proportional to the number of ounces in 1 lb. proportion always includes an equal sign,. hapter 7 182

2 Key oncept Properties of Proportions ross Products Property In a proportion a b 5 c, where b 2 0 and d 2 0, the d product of the etremes a and d equals the product of the means b and c. a b 5 c d a? d 5 b? 5 Equivalent Forms of Proportions a b 5 c d Property 1 Property 2 Property 3 is equivalent to a b 5 c d is equivalent to a b 5 c d b a 5 d c. a c 5 b d. a 1 b b 9. Identify the means and etremes in the proportion Means and Etremes and Identify the Property of Proportions each statement illustrates. 10. If , then is equivalent to 5 c 1 d. d 11. If , then 4(10) 5 5(8). 12. If , then If y, then y y. Problem 1 Write using the word to. to Writing a Ratio Got It? bonsai tree is 18 in. wide and stands 2 ft tall. What is the ratio of the width of the bonsai to its height? 14. The bonsai is in. wide and in. tall. 15. Write the same ratio three different ways. width of bonsai to height of bonsai Write as a fraction. Write using a colon. : 183 Lesson 7-1

3 Problem 3 Using an Etended Ratio Got It? The lengths of the sides of a triangle are in the etended ratio 4 : 7 : 9. The perimeter is 60 cm. What are the lengths of the sides? 16. Label the triangle at the right. Use the etended ratio to write an epression for each side length. 17. omplete the model to write an equation Relate the sum of the side lengths is the perimeter perimeter Write Use the justifications below to find the value of Write the equation.? 5 60 ombine like terms.? 5 60 Divide each side by. 5 Simplify. 19. Use the value of to find each side length ? 7 5 7? 5? The lengths of the sides of the triangle are cm, cm, and cm. Problem 4 Solving a Proportion Got It? lgebra What is the solution of the proportion a 14? 21. Write a justification for each statement below a 14 9(14) 5 2a a a 2 a 5 63 hapter 7 184

4 Problem 5 Writing Equivalent Proportions Got It? Use the proportion 6 5 y 7. What ratio completes the equivalent proportion 6 5 j j? Justify your answer. 22. Use the diagram at the right. Draw arrows from the and the 6 in the original proportion to the and the 6 in the new proportion. 23. ircle the proportion equivalent to a b 5 c that you can use. d 6 y 7 6 b a 5 d c a c 5 b d a 1 b b 5 c 1 d d 24. omplete: 6 5 y 7 is equivalent to 6 5. Lesson heck Do you UNDERSTND? Error nalysis What is the error in the solution of the proportion at the right? 25. ircle the means of the proportion. Then underline the etremes Write each product = = = 4 3 Means? 5 Etremes? What is the error in the solution of the proportion? 28. Now solve the proportion correctly. Math Success heck off the vocabulary words that you understand. proportion means etremes ross Products Property Rate how well you can solve proportions. Need to review Now I get it! 185 Lesson 7-1

5 7-2 Similar Polygons Vocabulary Review 1. What does it mean when two segments are congruent? 2. What does it mean when two angles are congruent? 3. Measure each segment. Then circle the congruent segments. Vocabulary uilder similar (adjective) SIM uh lur Other Word Forms: similarity (noun), similarly (adverb) Definition: Things that are similar are alike, but not identical. Math Usage: Figures that have the same shape but not necessarily the same size are similar. Use Your Vocabulary 4. How are the two squares at the right similar? 5. How are the two squares NOT similar? The symbol for similar is. hapter 7 186

6 Key oncept Similar Polygons Two polygons are similar polygons if corresponding angles are congruent and if the lengths of corresponding sides are proportional. D, GHIJ. Draw a line from each angle in olumn to its corresponding angle in olumn. olumn 6. / /H 7. / /J 8. / /G 9. /D /I olumn 10. omplete the etended proportion to show that corresponding sides of D and GHIJ are proportional. D H G GH D IJ I J Problem 1 Understanding Similarity Got It? DEFG, HJKL. What are the pairs of congruent angles? What is the etended proportion for the ratios of the lengths of corresponding sides? 11. omplete each congruence statement. 12. omplete the etended proportion. /D > / /E > / DE HJ 5 EF 5 KL 5 /K > / /L > / scale factor is the ratio of the lengths of corresponding sides of similar triangles. Problem 2 Determining Similarity Got It? re the polygons similar? If they are, write a similarity statement and give the scale factor. 13. ircle the short sides of each rectangle. Underline the long sides. KL LM MN NK WX XY YZ ZW 14. Write the ratios of corresponding sides in simplest form. KL XY LM YZ K 15 N 10 L W 20 MN ZW NK WX 5 5 X 15 M Z Y 187 Lesson 7-2

7 15. Place a in the bo if the statement is correct. Place an if it is incorrect. KLMN, XYZW and the scale factor is 2 3. KLMN, XYZW and the scale factor is 3 4. The polygons are not similar. Problem 3 Using Similar Polygons Got It? D, EFGD. What is the value of y? 16. ircle the side of D that corresponds to EF. D D 17. Use the justifications at the right to find the value of y. D EF ED 5 D orresponding sides of similar polygons are proportional. 9 6 E y 5 F G 7.5 y Substitute. 9y 5 ross Products Property y 5 Divide each side by 9. Problem 4 Using Similarity Got It? rectangular poster s design is 6 in. high by 10 in. wide. What are the dimensions of the largest complete poster that will fit in a space 3 ft high by 4 ft wide? 18. Determine how many times the design can be enlarged. Height: 3 ft 5 in. Width: 4 ft 5 in. in. 4 6 in. 5 6 in in The design can be enlarged at most times. 19. Let represent the height of the poster. Write a proportion and solve for. 20. The largest complete poster that will fit is in. by in. hapter 7 188

8 Problem 5 Using a Scale Drawing Got It? Use the scale drawing of the bridge. What is the actual height of the towers above the roadway? cm 21. Use a centimeter ruler to measure the height of the towers above the roadway in the scale drawing. Label the drawing with Scale 1 cm : 200 m the height. 22. Identify the variable. Let h 5 the 9 of the towers. 23. Use the information on the scale drawing to write a proportion. Then solve to find the value of the variable. QHint: tower height in drawing (cm) actual height (m) R 24. The actual height of the towers above the roadway is m. Lesson heck Do you UNDERSTND? The triangles at the right are similar. What are three similarity statements for the triangles? 25. The triangles are n and n. 26. / > / / > / /S > / 27. ns, ns, ns, Math Success heck off the vocabulary words that you understand. similar etended proportion scale factor scale drawing Rate how well you can identify and apply similar polygons. Need to review Now I get it! S R P 189 Lesson 7-2

9 7-3 Proving Triangles Similar Vocabulary Review Write the converse of each theorem. 1. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. If, then. 2. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If, then. Vocabulary uilder verify (verb) VEHR uh fy Related Word: proof (noun) Definition: To verify something means to find the truth or accuracy of it. Math Usage: proof is a way to verify a conjecture or statement. Use Your Vocabulary Write T for true or F for false. 3. You can verify that two triangles are similar by showing that corresponding angles are proportional. 4. You can use properties, postulates, and previously proven theorems to verify steps in a proof. hapter 7 190

10 Key oncept Postulate 7 1, Theorem 7 1, Theorem 7 2 Postulate 7-1 ngle-ngle Similarity (,) Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. Theorem 7-1 Side-ngle-Side Similarity (SS,) Theorem If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the triangles are similar. Theorem 7-2 Side-Side-Side Similarity (SSS,) Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar. 5. Write the postulate or theorem that proves the triangles similar Problem 1 Using the,postulate Got It? re the two triangles similar? How do you know? 6. omplete the diagram re the triangles similar? Eplain. Problem 2 Verifying Triangle Similarity Got It? re the triangles similar? If so, write a similarity statement for the triangles and eplain how you know the triangles are similar. 8. Write ratios for each pair of corresponding sides. 9. ircle the postulate or theorem you can use to verify that the triangles are similar., Postulate SS, Theorem SSS, Theorem 10. omplete the similarity statement G E 12 F n,n 191 Lesson 7-3

11 Problem 3 Proving Triangles Similar Got It? Given: uu MP Prove: n, npm P 11. The proof is shown below. Write a reason from the bo for each statement. ~ Postulate Given M Vertical angles are congruent. Statements Reasons 1) uu MP 1) 2) / > /P 2) If parallel lines are cut by a transversal, alternate interior angles are congruent. 3) / > /PM 3) 4) n, npm 4) Problem 4 Finding Lengths in Similar Triangles Got It? Reasoning Why is it important that the ground be flat to use the method of indirect measurement illustrated in the problem below? Eplain. efore rock climbing, Darius wants to know how high he will climb. He places a mirror on the ground and walks backward until he can see the top of the cliff in the mirror. 5.5 ft H T 6 ft V 12. If the ground is NOT flat, will /HTV and /JSV be right angles? Yes / No 13. If the ground is NOT flat, will you be able to find congruent angles? Yes / No 14. Why is it important that the ground be flat? Eplain. 34 ft J ft S hapter 7 192

12 Lesson heck Do you UNDERSTND? Error nalysis Which solution for the value of in the figure at the right is not correct? Eplain = 8 4 = 72 = = 6 48 = 4 12 = Write the side lengths of the triangles. Triangle Shortest Side Longest Side Third Side Larger 6 Smaller 16. Write ratios to compare the lengths of the corresponding sides. shortest sides: longest sides: third sides: 17. ross out the proportion that does NOT show ratios of corresponding sides Math Success Need to review Now I get it! ross out the solution that does NOT show ratios of corresponding sides. Solution Solution 19. Eplain why the solution you crossed out does NOT show the correct value of. heck off the vocabulary words that you understand. indirect measurement Rate how well you can prove triangles similar. similar triangles 193 Lesson 7-3

13 7-4 Similarity in Right Triangles Vocabulary Review Underline the correct word to complete the sentence. 1. The altitude of a triangle is a segment from a verte to the opposite side that is parallel / perpendicular to the opposite side. 2. In an isosceles triangle, the altitude to the base divides the triangle into two congruent / isosceles triangles. 3. ircle the altitude of n. D D Vocabulary uilder geometric mean (noun) jee uh MEH trik meen Definition: For any two positive numbers a and b, the geometric mean of a and b is the positive number such that a 5 b. Eample: The geometric mean of 4 and 10 is the value of in , or 5 2"10. Use Your Vocabulary 4. Multiple hoice Which proportion can you use to find the geometric mean of 5 and 15? Underline the correct equation to complete each sentence. 5. The geometric mean of a and b is 5 "ab / 5 ab. 6. The geometric mean of 3 and 7 is 5 "21 / ircle the geometric mean of!3 and! !3 3 3!3! hapter 7 194

14 Key oncept Theorem 7-3 and orollaries 1 and 2 Theorem 7-3 The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other. If... Then... n is a right triangle with right /, and D is the altitude to the hypotenuse D D D n, nd n, nd nd, nd orollary 1 to Theorem 7-3 If... Then... The length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse. D D D 5 D D orollary 2 to Theorem 7-3 If... Then... The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg. 8. nlmn is a right triangle with right /LMN. NP is the altitude to the hypotenuse. omplete the similarity statements. nlmn, n nlmn, n nlnp, n Use the triangle at the right. Write orollary 1 or orollary 2 for each proportion. 9. c a 5 a 10. m 5 m y 11. c b 5 b y D a m c y b 5 D 5 D L P N M 195 Lesson 7-4

15 Problem 1 Identifying Similar Triangles Got It? What similarity statement can you write relating the three triangles in the diagram? Q S R 12. Write the names of the triangles. nrpq n n P 13. Write the three right angles. 14. Write the three smallest angles. /RPQ / / /QRP / / 15. Use your answers to Eercises 13 and 14 to write three similarity statements beginning with the verte of the smallest angle in each triangle and ending with the verte of the right angle. nrqp, n nrqp, n n, n Problem 2 Finding the Geometric Mean Got It? What is the geometric mean of 4 and 18? 16. Use the justifications below to find the geometric mean. 4 5 Definition of geometric mean Å ross Products Property Take the positive square root of each side. 5 Å Problem 3 Using the orollaries Got It? What are the values of and y? Write in simplest radical form. Underline the correct word to complete each sentence is the length of a leg of the largest 18. y is the length of the altitude of the largest triangle, so use orollary 1 / orollary 2 triangle, so use orollary 1 / orollary 2 to find the value of. to find the value of y. 19. The values of and y are found below. Write a justification for each step y 4 y 5 y y "36 y 5 " y 5 2"5 hapter 7 196

16 Problem 4 Finding a Distance Got It? Points,, and are located so that 5 20 in., and '. Point D is located on so that D ' and D 5 9 in. You program a robot to move from to D and to pick up a plastic bottle at D. From point D, the robot must turn right and move to point to put the bottle in a recycling bin. How far does the robot travel from D to? 20. Place a in the bo if the statement is correct. Place an if it is incorrect. 20 in. 9 in. D 16 in. I know the length of the hypotenuse of n. I know the lengths of the segments of the hypotenuse of n. I know the length of the altitude of n. I can use orollary 1 to solve the problem. 21. Find the length of D. 22. The robot travels in. from D to. Lesson heck Do you UNDERSTND? Vocabulary Identify the following in nrst. 23. The hypotenuse is. 24. The segments of the hypotenuse are and. 25. The segment of the hypotenuse adjacent to leg ST is. Math Success heck off the vocabulary words that you understand. geometric mean altitude similarity Rate how well you understand similar right triangles. Need to review Now I get it! R S P T 197 Lesson 7-4

17 7-5 Proportions in Triangles Vocabulary Review 1. ircle the model that can form a proportion with ircle the ratios that you can use to form a proportion ross out the proportion that does NOT have the same solution as the others n 20 Vocabulary uilder 12 n n n bisector (noun) Y sek tur Other Word Form: bisect (verb) Definition: bisector divides a whole into two equal parts. Math Usage: bisector is a point, segment, ray, or line that divides an angle or a segment into two congruent angles or segments. Use Your Vocabulary Use the diagram at the right. omplete each statement with the correct word from the list below. Use each word only once. bisects bisector bisected 4. D ) is the 9 of /. 5. / is 9 by D ). D 6. D ) 9 /. hapter 7 198

18 Theorem 7-4 Side-Splitter Theorem and Its orollary Side-Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Q * ) * ) If RS uu XY, then XR RQ 5 SQ. 7. If XR 5 4, RQ 5 4, and YS 5 5, then SQ 5. X R S Y 8. If XR 5 3, RQ 5 6, and YS 5 4, then SQ 5. orollary to the Side-Splitter Theorem If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. If a 6 b 6 c, then 5 WX XY. omplete each proportion. a b c W X Y 9. 5 XY YX XW WX Problem 1 Using the Side-Splitter Theorem Got It? What is the value of a in the diagram at the right? 12. The value of a is found below. Use one of the reasons in the bo to justify each step. ross Products Property Divide each side by 6. Side-Splitter Theorem Simplify. Subtract 12a from each side. a a a 5 12a a 2 12a 5 12a 2 12a a a a 5 8 a 4 a Lesson 7-5

19 Problem 2 Finding a Length Got It? amping Three campsites are shown in the diagram. What is the length of Site along the road? 13. Let y be the length of Site along the road. Use the justifications at the right to find the value of y. y orollary to Side-Splitter Theorem? y ross Products Property? y Divide each side by the coefficient of y. 8 yd Site Site 9 yd 7.2 yd 6.4 yd Site y 5 Simplify. 14. The length of Site along the road is yd. Theorem 7-5 Triangle-ngle-isector Theorem Triangle-ngle-isector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. If D ) bisects /, then D D 5. D Problem 3 Using the Triangle-ngle-isector Theorem Got It? What is the value of y in the diagram at the right? 15. omplete the reasoning model below. Think I can use the Triangle-ngle-isector Theorem to write a proportion. hsm11_gemc_0705_t93531 Then I can use the ross-products Property. Now I divide each side by and simplify. 16. The value of y is. 16 Write y 16y y y y hapter 7 200

20 Lesson heck Do you know HOW? What is the value of in the figure at the right? 17. ircle the proportion you can use to solve the problem Solve the proportion. Lesson heck Do you UNDERSTND? Error nalysis classmate says you can use the Side-Splitter Theorem to find both and y in the diagram. Eplain what is wrong with your classmate s statement. 19. ross out the lengths that are NOT parts of the sides intersected by the parallel line y y 20. an you use the Side-Splitter Theorem to find? Yes / No 21. an you use the Side-Splitter Theorem to find y? Yes / No 22. Eplain what is wrong with your classmate s statement. Math Success heck off the vocabulary words that you understand. bisector proportion Side-Splitter Theorem Rate how well you understand side and angle bisectors. Need to review Now I get it! 201 Lesson 7-5