8.1 Similar Polygons Essential Question How are similar polygons related?
8.1 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.7.A G.7.B Similar Polygons Essential Question How are similar polygons related? Comparing Triangles after a Dilation Work with a partner. Use dynamic geometry software to draw any △ABC. Dilate △ABC to form a similar △A′B′C′ using any scale factor k and any center of dilation. B A C a. Compare the corresponding angles of △A′B′C′ and △ABC. b. Find the ratios of the lengths of the sides of △A′B′C′ to the lengths of the corresponding sides of △ABC. What do you observe? c. Repeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. Do you obtain similar results? Comparing Triangles after a Dilation ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, you need to look closely to discern a pattern or structure. Work with a partner. Use dynamic geometry software to draw any △ABC. Dilate △ABC to form a similar △A′B′C′ using any scale factor k and any center of dilation. B A a. Compare the perimeters of △A′B′C′ and △ABC. What do you observe? C b. Compare the areas of △A′B′C′ and △ABC. What do you observe? c. Repeat parts (a) and (b) for several other triangles, scale factors, and centers of dilation. Do you obtain similar results? Communicate Your Answer 3. How are similar polygons related? 4. A △RST is dilated by a scale factor of 3 to form △R′S′T′. The area of △RST is 1 square inch. What is the area of △R′S′T′? Section 8.1 Similar Polygons 421 What You Will Learn Lesson 8.1 Use similarity statements. Core Vocabul Vocabulary larry Find corresponding lengths in similar polygons. Previous similar figures similarity transformation corresponding parts Decide whether polygons are similar. Find perimeters and areas of similar polygons. Using Similarity Statements Recall from Section 4.6 that two geometric figures are similar figures if and only if there is a similarity transformation that maps one figure onto the other. Core Concept ANALYZING MATHEMATICAL RELATIONSHIPS Corresponding Parts of Similar Polygons In the diagram below, △ABC is similar to △DEF. You can write “△ABC is similar to △DEF ” as △ABC ∼ △DEF. A similarity transformation preserves angle measure. So, corresponding angles are congruent. A similarity transformation also enlarges or reduces side lengths by a scale factor k. So, corresponding side lengths are proportional. Notice that any two congruent figures are also similar. In △LMN and △WXY below, the scale factor is 5 6 7 —5 = —6 = —7 = 1. So, you can write △LMN ∼ △WXY and △LMN ≅ △WXY. M B a X 7 5 L E N 6 C 7 5 W 6 Y ka c b kc similarity transformation A F D kb Corresponding angles Ratios of corresponding side lengths ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F —=—=—=k DE AB EF BC FD CA Using Similarity Statements READING In the diagram, △RST ∼ △XYZ. In a statement of proportionality, any pair of ratios forms a true proportion. T Z a. Find the scale factor from △RST to △XYZ. b. List all pairs of congruent angles. 25 c. Write the ratios of the corresponding side lengths in a statement of proportionality. R 30 15 18 X 12 Y S 20 SOLUTION XY 12 3 a. — = — = — RS 20 5 YZ ST 18 30 3 5 —=—=— ZX TR 15 25 3 5 —=—=— 3 So, the scale factor is —. 5 b. ∠R ≅ ∠X, ∠S ≅ ∠Y, and ∠T ≅ ∠Z. XY YZ ZX c. Because the ratios in part (a) are equal, — = — = —. RS ST TR K 6 J 4 8 9 P 422 Chapter 8 Monitoring Progress Q L 6 12 Help in English and Spanish at BigIdeasMath.com 1. In the diagram, △JKL ∼ △PQR. Find the scale factor from △JKL to △PQR. R Similarity Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. Finding Corresponding Lengths in Similar Polygons Core Concept Corresponding Lengths in Similar Polygons READING Corresponding lengths in similar triangles include side lengths, altitudes, medians, and midsegments. If two polygons are similar, then the ratio of any two corresponding lengths in the polygons is equal to the scale factor of the similar polygons. Finding a Corresponding Length In the diagram, △DEF ∼ △MNP. Find the value of x. E SOLUTION The triangles are similar, so the corresponding side lengths are proportional. MN NP DE EF 18 30 —=— 15 x 18x = 450 —=— FORMULATING A PLAN There are several ways to write the proportion. For example, you could write EF DF — = —. MP NP x 15 Write proportion. D N Substitute. 30 18 Cross Products Property x = 25 F 20 Solve for x. M P 24 The value of x is 25. Finding a Corresponding Length In the diagram, △TPR ∼ △XPZ. —. Find the length of the altitude PS X T 6 S 6 R SOLUTION First, find the scale factor from △XPZ to △TPR. P 8 Y 8 20 TR 6 + 6 12 3 XZ 8 + 8 16 4 Because the ratio of the lengths of the altitudes in similar triangles is equal to the scale factor, you can write the following proportion. —=—=—=— PS 3 PY 4 PS 3 —=— 20 4 PS = 15 —=— Z Write proportion. Substitute 20 for PY. Multiply each side by 20 and simplify. — is 15. The length of the altitude PS Monitoring Progress 2. Find the value of x. A 12 10 D B x 16 C Help in English and Spanish at BigIdeasMath.com 3. Find KM. K Q 6 R 5 4 T 8 S G 40 H E 35 J ABCD ∼ QRST 48 M L F △JKL ∼ △EFG Section 8.1 Similar Polygons 423 Finding Perimeters and Areas of Similar Polygons Theorem ANALYZING MATHEMATICAL RELATIONSHIPS Theorem 8.1 Perimeters of Similar Polygons When two similar polygons have a scale factor of k, the ratio of their perimeters is equal to k. If two polygons are similar, then the ratio of their perimeters is equal to the ratios of their corresponding side lengths. K P L N M Q S R PQ + QR + RS + SP PQ QR RS SP If KLMN ∼ PQRS, then —— = — = — = — = —. KL + LM + MN + NK KL LM MN NK Proof Ex. 52, p. 430; BigIdeasMath.com Modeling with Mathematics A town plans to build a new swimming ppool. An Olympic pool is rectangular with a length of 50 meters and a width of w 225 meters. The new pool will be similar in sshape to an Olympic pool but will have a llength of 40 meters. Find the perimeters of aan Olympic pool and the new pool. 25 m 50 m SOLUTION S 11. Understand the Problem You are given the length and width of a rectangle and the length of a similar rectangle. You need to find the perimeters of both rectangles. 2. Make a Plan Find the scale factor of the similar rectangles and find the perimeter of an Olympic pool. Then use the Perimeters of Similar Polygons Theorem to write and solve a proportion to find the perimeter of the new pool. STUDY TIP You can also write the scale factor as a decimal. In Example 4, you can write the scale factor as 0.8 and multiply by 150 to get x = 0.8(150) = 120. 3. Solve the Problem Because the new pool will be similar to an Olympic pool, the 40 scale factor is the ratio of the lengths, — = —45 . The perimeter of an Olympic pool is 50 2(50) + 2(25) = 150 meters. Write and solve a proportion to find the perimeter x of the new pool. x 150 4 5 —=— Perimeters of Similar Polygons Theorem x = 120 Multiply each side by 150 and simplify. So, the perimeter of an Olympic pool is 150 meters, and the perimeter of the new pool is 120 meters. 4. Look Back Check that the ratio of the perimeters is equal to the scale factor. 120 150 4 5 —=— Gazebo B Gazebo A F 15 m G A 10 m B C x E 424 D ✓ 18 m 9m H 12 m K 15 m J Chapter 8 Similarity Monitoring Progress Help in English and Spanish at BigIdeasMath.com 4. The two gazebos shown are similar pentagons. Find the perimeter of Gazebo A. Theorem ANALYZING MATHEMATICAL RELATIONSHIPS When two similar polygons have a scale factor of k, the ratio of their areas is equal to k2. Theorem 8.2 Areas of Similar Polygons If two polygons are similar, then the ratio of their areas is equal to the squares of the ratios of their corresponding side lengths. Q P L K N M S R QR RS SP ). ( ) = ( LM ) = ( MN ) = ( NK Area of PQRS PQ If KLMN ∼ PQRS, then —— = — Area of KLMN KL 2 — 2 — 2 — 2 Proof Ex. 53, p. 430; BigIdeasMath.com Finding Areas of Similar Polygons In the diagram, △ABC ∼ △DEF. Find the area of △DEF. B E 10 cm 5 cm D A F C Area of △ABC = 36 cm2 SOLUTION Because the triangles are similar, the ratio of the area of △ABC to the area of △DEF is equal to the square of the ratio of AB to DE. Write and solve a proportion to find the area of △DEF. Let A represent the area of △DEF. 2 Area of △ABC AB Area of △DEF DE 2 36 10 —= — A 5 36 100 —=— A 25 36 25 = 100 A ( ) ( ) —— = — ⋅ ⋅ 900 = 100A 9=A Areas of Similar Polygons Theorem Substitute. Square the right side of the equation. Cross Products Property Simplify. Solve for A. The area of △DEF is 9 square centimeters. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. In the diagram, GHJK ∼ LMNP. Find the area of LMNP. P K 7m G J H L 21 m N M Area of GHJK = 84 m2 Section 8.1 Similar Polygons 425 Deciding Whether Polygons Are Similar Deciding Whether Polygons Are Similar Decide whether ABCDE and KLQRP are similar. Explain your reasoning. D 9 6 E C P 12 9 B 12 6 R 4 8 A Q 6 8 K L SOLUTION Corresponding sides of the pentagons are proportional with a scale factor of —23 . However, this does not necessarily mean the pentagons are similar. A dilation with center A and scale factor —23 moves ABCDE onto AFGHJ. Then a reflection moves AFGHJ onto KLMNP. D 9 6 C G 9 4 H 4 J 6 8 6 B 4 F E 8 A N P 6 R 8 K 4 M Q 6 8 L KLMNP does not exactly coincide with KLQRP, because not all the corresponding angles are congruent. (Only ∠A and ∠K are congruent.) Because angle measure is not preserved, the two pentagons are not similar. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Refer to the floor tile designs below. In each design, the red shape is a regular hexagon. Tile Design 1 Tile Design 2 6. Decide whether the hexagons in Tile Design 1 are similar. Explain. 7. Decide whether the hexagons in Tile Design 2 are similar. Explain. 426 Chapter 8 Similarity Exercises 8.1 Dynamic Solutions available at BigIdeasMath.com Vocabulary and Core Concept Check 1. COMPLETE THE SENTENCE For two figures to be similar, the corresponding angles must be ____________, and the corresponding side lengths must be _________________. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find “both” answers. What is the scale factor? A What is the ratio of their areas? 20 12 D 3 F C B 16 5 What is the ratio of their corresponding side lengths? E 4 △ABC ∼ △DEF What is the ratio of their perimeters? Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, find the scale factor. Then list all pairs of congruent angles and write the ratios of the corresponding side lengths in a statement of proportionality. (See Example 1.) 6. E x H 20 3. △ABC ∼ △LMN A 6.75 4.5 C D L B 6 9 6 7. N M 8 E R 1 Q 3 6 F 13 12 M 9 12 P x 26 24 L R S 4 Q 22 2 3 P 8. L 12 G J 12 J 6 K N 4. DEFG ∼ PQRS D G F 16 18 15 6 M G 4 H 9 15 In Exercises 5–8, the polygons are similar. Find the value of x. (See Example 2.) 5. P x 14 21 L 18 K J N x J 6 10 8 K P 20 Q 12 R Section 8.1 Similar Polygons 427 In Exercises 9 and 10, the black triangles are similar. Identify the type of segment shown in blue and find the value of the variable. (See Example 3.) 18. MODELING WITH MATHEMATICS Your family has decided to put a rectangular patio in your backyard, similar to the shape of your backyard. Your backyard has a length of 45 feet and a width of 20 feet. The length of your new patio is 18 feet. Find the perimeters of your backyard and of the patio. 9. 27 x In Exercises 19–22, the polygons are similar. The area of one polygon is given. Find the area of the other polygon. (See Example 5.) 16 18 19. 10. 3 ft 18 y 6 ft A = 27 16 20. y−1 ft2 4 cm 12 cm A = 10 cm2 In Exercises 11 and 12, RSTU ∼ ABCD. Find the ratio of their perimeters. 11. R 21. S A B 12 U D T 4 in. 8 20 in. A = 100 in.2 C 14 22. 12. R 18 A S 24 B 3 cm U 36 T D 12 cm A = 96 cm2 C In Exercises 13–16, two polygons are similar. The perimeter of one polygon and the ratio of the corresponding side lengths are given. Find the perimeter of the other polygon. 2 13. perimeter of smaller polygon: 48 cm; ratio: —3 23. ERROR ANALYSIS Describe and correct the error in finding the perimeter of triangle B. The triangles are similar. ✗ 10 A 5 1 15. perimeter of larger polygon: 120 yd; ratio: —6 2 16. perimeter of larger polygon: 85 m; ratio: —5 17. MODELING WITH MATHEMATICS A school gymnasium is being remodeled. The basketball court will be similar to an NCAA basketball court, which has a length of 94 feet and a width of 50 feet. The school plans to make the width of the new court 45 feet. Find the perimeters of an NCAA court and of the new court in the school. (See Example 4.) B 24. ERROR ANALYSIS Describe and correct the error in finding the area of rectangle B. The rectangles are similar. ✗ A = 24 units2 A 6 B 18 428 Chapter 8 Similarity 5x = 280 x = 56 12 3 14. perimeter of smaller polygon: 66 ft; ratio: —4 28 5 =— — 10 x 6 6 24 =— — 18 x 6x = 432 x = 72 In Exercises 25 and 26, decide whether the red and blue polygons are similar. (See Example 6.) 25. 36. DRAWING CONCLUSIONS In table tennis, the table is a rectangle 9 feet long and 5 feet wide. A tennis court is a rectangle 78 feet long and 36 feet wide. Are the two surfaces similar? Explain. If so, find the scale factor of the tennis court to the table. 40 30 22.5 30 26. 3 3 3 3 3 3 MATHEMATICAL CONNECTIONS In Exercises 37 and 38, the two polygons are similar. Find the values of x and y. 27. REASONING Triangles ABC and DEF are similar. 37. 27 Which statement is correct? Select all that apply. BC AC AB CA A —=— ○ EF DF B —=— ○ DE FE BC D —=— ○ FD EF AB CA C —=— ○ EF DE 39 24 BC ANALYZING RELATIONSHIPS In Exercises 28 –34, y x 38. JKLM ∼ EFGH. (y − 73)° J H3G y 11 z° E 8 F 20 116° 116° 5 M x K L 61° 4 6 30 65° 18 x−6 ATTENDING TO PRECISION In Exercises 39– 42, the 28. Find the scale factor of JKLM to EFGH. figures are similar. Find the missing corresponding side length. 29. Find the scale factor of EFGH to JKLM. 39. Figure A has a perimeter of 72 meters and one of the side lengths is 18 meters. Figure B has a perimeter of 120 meters. 30. Find the values of x, y, and z. 31. Find the perimeter of each polygon. 40. Figure A has a perimeter of 24 inches. Figure B 32. Find the ratio of the perimeters of JKLM to EFGH. 33. Find the area of each polygon. has a perimeter of 36 inches and one of the side lengths is 12 inches. 41. Figure A has an area of 48 square feet and one of 34. Find the ratio of the areas of JKLM to EFGH. the side lengths is 6 feet. Figure B has an area of 75 square feet. 35. USING STRUCTURE Rectangle A is similar to 42. Figure A has an area of 18 square feet. Figure B rectangle B. Rectangle A has side lengths of 6 and 12. Rectangle B has a side length of 18. What are the possible values for the length of the other side of rectangle B? Select all that apply. A 6 ○ B 9 ○ C 24 ○ has an area of 98 square feet and one of the side lengths is 14 feet. D 36 ○ Section 8.1 Similar Polygons 429 CRITICAL THINKING In Exercises 43 –48, tell whether the polygons are always, sometimes, or never similar. 52. PROVING A THEOREM Prove the Perimeters of Similar Polygons Theorem (Theorem 8.1) for similar rectangles. Include a diagram in your proof. 43. two isosceles triangles 44. two isosceles trapezoids 45. two rhombuses 53. PROVING A THEOREM Prove the Areas of Similar 46. two squares Polygons Theorem (Theorem 8.2) for similar rectangles. Include a diagram in your proof. 47. two regular polygons 48. a right triangle and an equilateral triangle 54. THOUGHT PROVOKING The postulates and theorems in this book represent Euclidean geometry. In spherical geometry, all points are points on the surface of a sphere. A line is a circle on the sphere whose diameter is equal to the diameter of the sphere. A plane is the surface of the sphere. In spherical geometry, is it possible that two triangles are similar but not congruent? Explain your reasoning. 49. MAKING AN ARGUMENT Your sister claims that when the side lengths of two rectangles are proportional, the two rectangles must be similar. Is she correct? Explain your reasoning. 50. HOW DO YOU SEE IT? You shine a flashlight directly on an object to project its image onto a parallel screen. Will the object and the image be similar? Explain your reasoning. 55. CRITICAL THINKING In the diagram, PQRS is a square, and PLMS ∼ LMRQ. Find the exact value of x. This value is called the golden ratio. Golden rectangles have their length and width in this ratio. Show that the similar rectangles in the diagram are golden rectangles. P S 51. MODELING WITH MATHEMATICS During a total 1 x Q L R M 56. MATHEMATICAL CONNECTIONS The equations of the eclipse of the Sun, the moon is directly in line with the Sun and blocks the Sun’s rays. The distance DA between Earth and the Sun is 93,000,000 miles, the distance DE between Earth and the moon is 240,000 miles, and the radius AB of the Sun is 432,500 miles. Use the diagram and the given measurements to estimate the radius EC of the moon. lines shown are y = —43 x + 4 and y = —43 x − 8. Show that △AOB ∼ △COD. y B A O B C x C D A Sun E moon D Earth △DEC ∼ △DAB △ Not drawn to scale Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the value of x. (Section 5.1) 57. 58. 59. x° x° 76° 24° 60. 52° 41° x° 430 Chapter 8 Similarity x°