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M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY ASSIGNMENTS FOR PACKET 1 OF UNIT 5 This packet includes sections 7-1, 7-2, and 7-3 from our textbook. Date Due Number Assignment 5A p. 464 # 1 – 4 all p. 465 # 34, 37, 40 5B p. 465 # 23 – 26 all, 29 – 31 all 5C p. 473-474 # 9, 11, 12, 14, 19, 21, 23, 28 5D p. 483-484 # 1 – 4 all, 7, 8, 16, 17, 18, 22 HINT: In #2 & #18, use the Pythagorean Theorem a2 + b2 = c2 to find the 3rd side of a right triangle if you know the other two. 1 Topics 7-1: Vocabulary: ratio, proportion, cross products Use ratios to solve problems Use ratios and algebra to solve problems. 7-1: Vocabulary: proportion, cross products Use algebra to solve proportions involving linear or quadratic equations. Use proportions to solve real world problems. 7-2: Vocabulary: similar polygons, scale factor Name and identify corresponding parts of similar polygons Use proportions to solve problems involving similar polygons 7-3: Use SSS, SAS, or AA Similarity to determine whether two triangles are similar Use proportions to solve problems involving similar triangles M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY 7-1 Ratios and Proportions Part 1: Ratios A ratio compares two quantities using division. The ratio a to b can be written as a or a:b. b Ex. 1: In 2015 the Mets won 90 games out of 162 games played. Write a ratio for the number of games won to the total number of games played. Give your answer as a reduced fraction and as a percent rounded to the nearest tenth. Solution: To find the ratio, divide number of games won by number of games played. The result is Using Math, Frac on a calculator to reduce, we get 90 . 162 5 . 9 As a decimal, 5/9 is 0.556, which means 55.6%. Try these: 1. In 2016, Kris Bryant hit 39 home runs and was at bat 603 times. What is the ratio of home runs to the number of times he was at bat? Give your answer as a reduced fraction and as a percent rounded to the nearest tenth. 2. Suppose there are 182 girls in a sophomore class of 306 students. What is the ratio of girls to total students? Give your answer as a reduced fraction and as a percent rounded to the nearest tenth. 2 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Part 2: Extended Ratios Ex. 2: The ratio of the measures of the angles in JKH is 2:3:4. Find the measures of the angles. Solution: This is called an extended ratio because it has more than two parts. 2:3:4 can be rewritten as 2x:3x:4x. Sketch and label the angle measures of the triangle as shown at right. Then write and solve an equation to find the value of x. 2x + 3x + 4x = 180 9x = 180 x = 20 The angle measures are 2(20) = 40 , 3(20) = 60 , and 4(20) = 80 . Try these: 3. The ratio of the sides of a triangle is 8:15:17. Its perimeter is 480 inches. Find the length of each side. 4. The ratio of the measures of angles in a triangle is 7:9:20. Find the measure of each angle. 3 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Part 3: Solving Basic Proportions An equation stating that two ratios are equal is called a proportion. In the proportion a c , the cross products are equal. That is, ad bc . b d 9 27 . 16 x Solution: 9 x 16 27 9 x 432 x 48 Ex. 3: Solve Try these: Solve each proportion. Show all steps. 5. 9 3 y 18.2 x3 8 . 3 x2 Solution: x 3 x 2 3 8 6. x 22 30 x 2 10 8. 4 x5 x 3 5 Ex. 4: Solve x2 2 x 3x 6 24 x2 x 6 24 x2 x 30 0 x 6 x 5 0 x 6 0 or x 5 0 x 6 or x 5 Cross products FOIL Simplify Get 0 on one side. Factor Set factors = to 0. Solve each for x. Try these: Solve each proportion. Show all steps. 7. x5 6 5 x2 4 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Part 4: Using Proportions to Solve Word Problems Ex. 5: The mayor conducted a random survey of 200 voters and found that 135 approve of the job he is doing. If there are 48,000 voters in this district, predict the total number of voters who approve of the job he is doing. Solution: Surveyed voters who approve All surveyed voters 135 x 200 48, 000 District voters who approve All district voters 135 48,000 200x 6, 480,000 200x 32, 400 x Based on the survey, about 32,400 registered voters approve of the job the mayor is doing. Try these: 9. Mark is measuring plants in a field. Of the first 25 plants he measures, 15 are smaller than a foot in height. If there are 4000 plants in the field, predict the total number of plants smaller than a foot in height. 10. A storm produced a rainfall of 2 inches in one hour. At this rate, how many hours would it take to get a rainfall amount of one foot? 5 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY 7-2 Similar Polygons Polygons are similar if: (1) Corresponding angles are congruent, and (2) Corresponding sides are in the same proportion. The statement XYZ means that triangle ABC is similar to triangle XYZ. ABC Ex. 1: If ABC XYZ , list all pairs of congruent angles, and write a proportion that relates the corresponding sides. Solution: Use the order of the letters in the triangle names: Congruent angles: A X , B Y , C Z Proportion: AB BC CA XY YZ ZX 1. Try this: List all pairs of congruent angles, and write a proportion that relates the corresponding sides. PQRS TUWX Congruent angles: ____ ____ , ____ ____ , ____ ____ , ____ ____ Proportion: Conversely, if two polygons have congruent corresponding angles and proportional corresponding sides, then they are similar. Ex. 2: Determine whether the figures are similar. If so, write a similarity statement, and find the scale factor. Solution: Compare corresponding angles. W P, X Q, Y R, Z S Corresponding angles are congruent. Compare corresponding sides by writing ratios for each pair. WX 12 3 PQ 8 2 XY 18 3 QR 12 2 YZ 15 3 RS 10 2 ZW 9 3 SP 6 2 Corresponding sides are proportional because all of the ratios are equal. The scale factor is 3 , and WXYZ 2 PQRS . 6 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Try these: Determine whether the figures are similar. If so, write a similarity statement, and find the scale factor. If not, explain your reasoning. 2. 3. Ex. 3: The two polygons are similar. Find x and y. Solution: Use the congruent angles to write corresponding vertices in order: RST MNP Write and solve proportions: RS ST MN NP 32 x 16 13 RS RT MN MP 32 38 16 y 16 x 32 13 32 y 38 16 y 19 x 26 Try these: Each pair of polygons is similar. Find the value of x. 4. 5. 7 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY If two polygons are similar, the ratio of their perimeters will equal the ratio of their sides. We can use this fact to find the perimeter of a figure without finding the lengths of all of its sides. Ex. 4: If DEF GHJ , find the scale factor of and the perimeter of each triangle. Solution: The scale factor is The perimeter of Then DEF to GHJ EF 8 2 . HJ 12 3 DEF is 10 + 8 + 12 = 30. 2 perimeter of DEF . 3 perimeter of GHJ 2 30 3 x 2 x 3 30 x 45 6. Try this: If ABCD The perimeter of GHJ is 45. PQRS , find the scale factor of ABCD to PQRS and the perimeter of each polygon. 8 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY 7-3 Similar Triangles Here are three ways to show that two triangles are similar. AA Similarity SSS Similarity SAS Similarity Two angles of one triangle are congruent to two angles of another triangle. Lengths of all three pairs of corresponding sides of two triangles are proportional. Two side lengths of one triangle are proportional to the two corresponding side lengths of another triangle, and the included angles are congruent. Ex. 1 : Determine whether the triangles are similar. a. b. Solution: Solution: AC 6 2 BC 8 2 AB 10 2 , , and DF 9 3 EF 12 3 DE 15 3 ABC DEF by SSS Similarity. MN NP 3 6 , so QR RS 4 8 Also, N R . NMP RQS by SAS Similarity. Try these: Determine whether the triangles are similar. If so, write a similarity statement. Explain your reasoning. 1. 2. 3. 9 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Ex. 2 : Write a similarity statement, and explain why the triangles are similar. Then find IU. Solution: G G . Also, GWH Since GW 26 2 GH 26 2 and . GU 39 3 GI 39 3 GUI by SAS Similarity. GW HW 26 20 , then . So 26 x 39 20 , or x 30 . GU IU 39 x Try these: Write a similarity statement, and explain why triangles are similar. Then find the requested measure. 4. JL 5. BC Ex. 3 : A person 6 feet tall casts a 1.5-foot-long shadow at the same time that a flagpole casts a 7-foot-long shadow. How tall is the flagpole? Solution: The Sun’s rays form similar triangles. Using x for the height of the pole, 1.5 6 . 7 x So 1.5 x 6 7 , and x 28 . The flagpole is 28 feet tall. 6. Try this: Two vertical posts are 2 meters and 0.45 meter tall. When the shorter post casts a shadow that is 0.85 meter long, what is the length of the longer post’s shadow? Round your answer to the nearest hundredth. 10 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Practice Problems for 7-1 to 7-3 #1-3: Find the requested ratio. (See page 2 of this packet.) 1. Kieran scored 6 touchdowns in 14 games. Find the ratio of touchdowns per game. Express as a reduced fraction. 2. In a schedule of 6 classes, Katie has 2 elective classes. What is the ratio of elective to non–elective classes in Katie’s schedule? Express as a reduced fraction. 3. Out of 274 listed species of birds in the United States, 78 species made the endangered list. Find the ratio of endangered species of birds to listed species in the United States. Express as percent rounded to the nearest tenth. #4-5: Write an equation, and solve. (See page 3 of this packet.) 4. The ratio of male students to female students in the Latin club at Campbell High School is 3:4. If there are 18 male students in the club, how many female students are in the club? 5. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters. What are the measures of the sides of the triangle? #6-8: Solve each proportion. (See page 4 of this packet.) 6. 20 4 x 5 6 7. x 1 7 3 2 8. 11 15 x 3 3 5 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY #9-10: Solve each proportion. (See page 5 of this packet.) 9. x3 11 4 x4 10. x 5 4 8 x 1 #11-12: Determine whether the figures are similar. If so, write the similarity statement, and find the scale factor. If not, explain your reasoning. (See pages 6-7 of this packet.) 11. 12. #13-14: Each pair of polygons is similar. Write a proportion, and solve for x. (See pages 5-7 of this packet.) 13. 14. + 12 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY #15-18: Determine whether each pair of triangles is similar. If so, write a similarity statement. Explain your reasoning. (See page 9 of this packet.) 15. 16. 17. 18. #19-21: Write a similarity statement, and explain why the triangles are similar. Then find the requested measure. (See page 10 of this packet.) 19. AC 20. JL 21. VT 13 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Review for 7-1 to 7-3 1. One ounce of cheddar cheese contains 9 grams of fat. Six of the grams of fat are saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese. Write your answer as a reduced fraction and as a percent rounded to the nearest tenth. 2. Write a proportion and solve: A worker at an automobile assembly plant checks new cars for defects. Of the first 280 cars he checks, 4 have defects. If 10,500 cars will be checked this month, predict the total number of cars that will have defects. 3. Write an equation and solve: The ratio of goats to sheep at a university research farm is 4:7. The number of sheep at the farm is 28. What is the number of goats? 4. Write an equation and solve: The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet. Find the measure of each side of the triangle. 5. Write an equation and solve: The ratio of the measures of the three angles is 4:5:6. Find the measure of each angle of the triangle. Solve each proportion. 6. 3x 5 5 4 7 7. 4 2 x2 x4 8. 2 x5 x4 3 9. 5 x4 x 1 10 14 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Determine whether each pair of figures is similar. If so, write the similarity statement, and find the scale factor. If not, explain your reasoning. 10. 11. Each pair of polygons is similar. Find the value of x. 12. 13. 14. If ABCDE PQRST , find the scale factor of ABCDE to PQRST and the perimeter of each polygon. Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Explain your reasoning. 15. 16. 15 M2 GEOMETRY PACKET 1 FOR UNIT 5 – SECTIONS 7-1 TO 7-3 INTRODUCTION TO SIMILARITY Write a similarity statement, and explain why the triangles are similar. Then find the requested measure. 17. LM, QP 18. NL, ML 19. A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow. a. Sketch a diagram, and label with the given information. b. Write a proportion that can be used to determine the height of the lighthouse. c. Find the height of the lighthouse. Answers to Review Problems on p. 14-16 in this packet: 4 x 1. 2/3, 66.7% 2. , 150 cars 280 10,500 3. 7x = 28, 16 goats 4. 3x + 4x + 6x = 104, 24 ft, 32 ft, 48 ft 5. 4x + 5x + 6x = 180, 48 , 60 , 72 6. x = 5/7 7. x = 10 8. x = 7 or x = 2 9. x = 9 or x = 6 ABC UVT , scale factor 2/3 10. MLKJ PSQR, scale factor 5/3 11. 12. x = 1.5 13. x = 7 14. scale factor 5/7, perim of ABCDE = 65, perim of PQRST = 91 SWY by SAS Similarity 15. AJK 16. No. We need to know that the 3rd pair of sides has the same ratio or that LNM Q . 17. PQN by AA Similarity, x = 9 LMN 19. b. x 5.25 128 8 18. c. 84 ft 16 KLM JLN by AA Similarity, x = 2