Similar Shapes and Proportions 7.5.A

Transcription

1 LESSON 4.1 Similar Shapes and Proportions Proportionality Generalize the critical attributes of similarity, including ratios within and between similar shapes.? ESSENTIL QUESTION How can you use ratios to determine if two figures are similar? EXPLORE TIVITY Similar Shapes and Proportions Similar shapes have the same shape but not necessarily the same size. You can use square tiles to model similar figures. rectangle made of square tiles measures 5 tiles long and tiles wide. ind the length of a similar rectangle that measures 6 tiles wide. 5 STEP 1 STEP Using the square tiles, make a rectangle 5 tiles long and tiles wide. dd tiles to increase the width of the rectangle to 6 tiles. There are now sets of the original tiles along the width of the rectangle because = 6. The width of the rectangle is original rectangle. times the width of the STEP 3 dd tiles to also increase the length of the rectangle. The width of the new rectangle is times the width of the original. To keep the lengths of the sides proportional, the length must also be times the length of the original. The length of the similar rectangle is = 15. The length of the similar rectangle is tiles. Reflect 1. Justify Reasoning If one dimension is changed, why does the other dimension have to change to create a similar figure? Lesson

2 Math On the Spot etermining Whether Two Triangles re Similar Similar shapes have the same shape, but not necessarily the same size. orresponding angles and corresponding sides of two or more similar shapes are in the same relative position. The symbol ~ means is similar to. In the figure shown, sides and angles that are the same color correspond to each other. B ~ E. B E Similar igures In two similar figures: the measures of their corresponding angles are equal, and the lengths of their corresponding sides are proportional. EXMPLE 1 Math Talk Mathematical Processes re all equilateral triangles similar? Explain. Explain whether the triangles are similar. STEP 1 STEP heck that the corresponding angles of the triangles have equal measures. m E = m R = 106 m = m S = 34 m = m Q = 40 E corresponds to QR. corresponds to QS. E?? QR = E RS = QS 7?? 1 = 8 4 = ?? 3 = 1 3 = in. 8 in. 40 E 7 in. 106 E corresponds to R corresponds to S corresponds to Q S 34 4 in. 36 in R 1 in. heck that the corresponding side lengths are proportional. E corresponds to RS. Write ratios using the lengths of corresponding sides. Substitute the lengths of the sides. Simplify. Since the measures of the corresponding angles are equal and the corresponding sides are proportional, the triangles are similar. Q side of a figure can be named by its endpoints with a bar above. 116 Unit

3 YOUR TURN Explain whether the triangles are similar ft 5 ft 17 ft ft B 11 ft E ft in in in. B E 16 in in. Personal Math Trainer Online ssessment and Intervention in. 18 etermining Whether Two our-sided igures re Similar Shapes with four or more sides can also be similar if the corresponding angles have equal measures and the corresponding side lengths are proportional. EXMPLE iana makes two sizes of earrings. Explain whether the shapes of the earrings are similar. STEP 1 STEP Note that each ratio in simplest form is 5 3. O S 10 mm 18 mm 6 mm 10.8 mm N P R T mm 6 mm 15 mm M 86 Q 9 mm heck that the corresponding angles have equal measures. The angle measures are 48, 86, 101, and 15 in both earrings. heck that the corresponding side lengths are proportional. MN corresponds to QR OP corresponds to ST MN QR =? NO RS =? OP ST =? PM TQ 10 6 =? 10 6 =? =? 15 9 NO corresponds to RS PM corresponds to TQ Write ratios using the lengths of corresponding sides. Substitute the lengths of the sides. Math On the Spot nimated Math 1. 6 = 1. 6 = 1. 6 = 1. 6 ivide. The measures of the corresponding angles are equal and the corresponding sides are proportional, so the earrings are similar. Lesson

4 YOUR TURN Personal Math Trainer Online ssessment and Intervention Explain whether the shapes are similar. 4. rectangle B with sides of 7 and 5 and rectangle MNOP with sides of 1 and ft 6 ft 6 ft ft ft ft 18 ft ft Guided Practice 1. rectangle made of square tiles measures 7 tiles long and 3 tiles wide. What is the length of a similar rectangle whose width is 9 tiles? (Explore ctivity) Explain whether the shapes are similar. (Examples 1 and )?. Q K cm 70 J 1 cm 18 cm 70 L 48 cm 48 cm 73 P 8 cm 73 R ESSENTIL QUESTION HEK-IN 4. escribe how to use ratios to determine whether two shapes are similar. 1 ft ft ft 1 ft 14 ft 14 ft ft 1 ft 118 Unit

5 Name lass ate 4.1 Independent Practice Personal Math Trainer Online ssessment and Intervention etermine if each statement is true or false. Justify your answer. 5. ll squares are similar. 6. ll right triangles are similar. rt or 7 10, use the table. ssume all angle measures are equal to Hugo has a small print of one of the paintings in the table. It is similar in size to the original. The print measures 11 in. 10 in. Of which painting is this a print? Explain. Painting Mona Lisa The ance lass rtist Leonardo da Vinci Edgar egas Original Size (in.) 30 x 1 33 x 30 The Blue Vase Paul ézanne 8 x 18 Image redits: omstock/ Getty Images/Houghton Mifflin Harcourt 8. local artist painted a copy of ezanne s painting. It measures 88 in. 74 in. Is the copy similar to the original? Explain. 9. company made a poster of da Vinci s painting. The poster is 5 feet long and 3.5 feet wide. Is the poster similar to the original Mona Lisa? Explain. 10. The same company made a poster of The Blue Vase. The poster is 36 inches long and 6 inches wide. Is the poster similar to the original The Blue Vase? Explain. Lesson

6 Problem Solving The figure shows a 1 ft by 15 ft garden divided into four rectangular parts, each planted with a different vegetable. Explain whether the rectangles in each pair are similar and why. 1 ft 5 ft B 11. rectangle and the original rectangle 4 ft 15 ft 1. the original rectangle and rectangle 13. rectangle and rectangle B OUS ON HIGHER ORER THINKING Work rea 14. nalyze Relationships Which of these four-sided shapes are similar? Q T R S 4 cm 8 cm 8 cm B 4 cm Z W Y X 15. ommunicate Mathematical Ideas escribe the two tests two polygons must pass to be proven similar. 16. Make a onjecture Using what you know of similar figures, explain whether you believe all rectangles are similar. Give an example or a counterexample. 10 Unit