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Areas of Similar Polygons and Scale Models

Learn how to find the areas of similar polygons and use scale factors to determine dimensions in scale models.

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Areas of Similar Polygons and Scale Models

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  1. Splash Screen

  2. Five-Minute Check (over Lesson 11–4) NGSSS Then/Now Theorem 11.1: Areas of Similar Polygons Example 1: Find Areas of Similar Polygons Example 2: Use Areas of Similar Figures Example 3: Real-World Example: Scale Models Lesson Menu

  3. A B C D What is the area of a regular hexagon with side length of 8 centimeters? Round to the nearest tenth if necessary. A. 48 cm2 B. 144 cm2 C. 166.3 cm2 D. 182.4 cm2 5-Minute Check 1

  4. A B C D What is the area of a square with an apothem length of 14 inches? Round to the nearest tenth if necessary. A. 784 in2 B. 676 in2 C. 400 in2 D. 196 in2 5-Minute Check 2

  5. A B C D Find the area of the figure. Round to the nearest tenth if necessary. A. 120 units2 B. 114 units2 C. 108 units2 D. 96 units2 5-Minute Check 3

  6. A B C D Find the area of the figure. Round to the nearest tenth if necessary. A. 184 units2 B. 158.9 units2 C. 132.6 units2 D. 117.7 units2 5-Minute Check 4

  7. A B C D Find the area of the figure. A. 11 units2 B. 12 units2 C. 13 units2 D. 14 units2 5-Minute Check 5

  8. A B C D Find the area of a regular triangle with a side length of 18.6 meters. A. 346 m2 B. 299.6 m2 C. 173 m2 D. 149.8 m2 5-Minute Check 6

  9. MA.912.G.2.6Use coordinate geometry to prove properties of congruent, regular and similar polygons, and to perform transformations in the plane. MA.912.G.2.7 Determine how changes in dimensions affect the perimeter and area of common geometric figures. NGSSS

  10. You used scale factors and proportions to solve problems involving the perimeters of similar figures. (Lesson 7–2) • Find areas of similar figures by using scale factors. • Find scale factors or missing measures given the areas of similar figures. Then/Now

  11. Concept

  12. 9 3 __ __ 6 2 The scale factor between PQRS and ABCD is or .So, the ratio of the areas is Find Areas of Similar Polygons If ABCD ~ PQRS and the area of ABCD is 48 square inches, find the area of PQRS. Example 1

  13. Find Areas of Similar Polygons Write a proportion. Multiply each side by 48. Simplify. Answer: So, the area of PQRS is 108 square inches. Example 1

  14. A B C D If EFGH ~ LMNO and the area of EFGH is 40 square inches, find the area of LMNO. A. 180 ft2 B. 270 ft2 C. 360 ft2 D. 420 ft2 Example 1

  15. Use Areas of Similar Figures The area of ΔABC is 98 square inches. The area of ΔRTS is 50 square inches. If ΔABC ~ ΔRTS, find the scale factor from ΔABC to ΔRTS and the value of x. Let k be the scale factor between ΔABC and ΔRTS. Example 2

  16. So, the scale factor from ΔABC to ΔRTS isUse the scale factor to find the value of x. Use Areas of Similar Figures Theorem 11.1 Substitution Simplify. Take the positive square root of each side. Example 2

  17. Use Areas of Similar Figures The ratio of corresponding lengths of similar polygons is equal to the scale factor between the polygons. Substitution 7x = 14 ● 5 Cross Products Property. 7x = 70 Multiply. x = 10 Divide each side by 7. Answer: x = 10 Example 2

  18. CHECK Confirm that is equal to the scale factor. Use Areas of Similar Figures  Example 2

  19. A B C D The area of ΔTUV is 72 square inches. The area ofΔNOP is 32 square inches. If ΔTUV ~ ΔNOP, use the scale factor from ΔTUV to ΔNOP to find the value of x. A. 3 inches B. 4 inches C. 6 inches D. 12 inches Example 2

  20. Scale Models CRAFTS The area of one side of a skyscraper is 90,000 square feet. The area of one side of a scale model is 200 square inches. If the skyscraper is 720 feet tall, about how tall is the model? Understand The sides of the skyscraper and the scale model are similar. You need to find the scale factor from the skyscraper to the scale model. Plan The ratio of the areas of the sides of the two figures is equal to the square of the scale factor between them. Before comparing the two areas, write them so that they have the same units. Example 3

  21. Scale Models Solve Convert the areas of the scale model to square feet. 1.389 ft2 Next, write an equation using the ratio of the two areas in square feet. Let k represent the scale factor between the two sides. Example 3

  22. Scale Models Theorem 11.1 Substitution 1.54 ● 10–5 = k2 Simplify using a calculator. Example 3

  23. So, the model’s height is the height of the skyscraper. Multiply the height of the skyscraper by the scale factor and convert to inches. Scale Models Answer: The scale model is about 34 inches tall. Example 3

  24. Scale Models CHECKMultiply the area of the side of skyscraper by the square of this scale factor and compare to the given area of the side of the scale model.  Example 3

  25. A B C D MODELS The area of one hood of a car is 35 square feet. The area of the hood of a model is 6 square inches. If the car is 14 feet long, about how long is the model? A. 4.3 inches B. 5.8 inches C. 6.7 inches D. 7.2 inches Example 3

  26. End of the Lesson

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