468 CHAPTER 8 Geometry Did You Know? Buckyballs m The molecular structure of C60 resembles the patterns found on a soccer ball. Buckminsterfullerenes, known as buckyballs, are a class of pure carbon molecules. Along with graphite and diamond, buckyballs are the only naturally occurring forms of pure carbon. Named after American architect–engineer F. Buckminster Fuller, who designed hemispherical geodesic domes from hexagonal and pentagonal faces, fullerenes are the most spherical molecules known. Discovered in 1985 at Rice University, buckyballs are used primarily as microscopic lubricant and have potential applications in molecular medical engineering, electrical superconductivity, and computer chip design. The most common form of buckminsterfullerene contains 60 carbon atoms and has the chemical symbol C . 60 The molecular structure of C60 contains 12 pentagons and 20 hexagons arranged in a pattern similar to that found on a soccer ball. Example 2 Similar Figures Consider the similar figures in Fig. 8.21. A B E C D 3 4 x Figure 8.21 P Q T S R 4.5 3 y Determine a) the length of side BC. b) the length of side PQ. Solution a) We will represent the length of side BC with the variable x. Because the corresponding sides of similar figures must be in proportion, we can write a proportion (as explained in Section 6.3) to determine the length of side BC. The lengths of corresponding sides CD and RS are known, so we use them as one ratio in the proportion. The side corresponding to BC is QR. = = BC QR CD RS x 3 3 4.5 Now we solve for x. ⋅ = ⋅ = = x x x 4.5 3 3 4.5 9 2 Cross multiply Thus, the length of side BC is 2 units. b) We will represent the length of side PQ with the variable y. The side corresponding to PQ is AB. We will work part (b) in a manner similar to part (a). = = ⋅ = ⋅ = = PQ AB RS CD y y y y 4 4.5 3 3 4 4.5 3 18 6 Thus, the length of side PQ is 6 units. 7 Now try Exercise 49 Example 3 Determine the Height of a Tree Allisha wants to know the height of a tree in her back yard. Allisha is 6 ft tall and determines that when her shadow is 9 ft long, the shadow of the tree is 45 ft long (see Fig. 8.22). How tall is the tree? Keanu/Shutterstock
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