8.2 Polygons 467 Similar Figures In everyday living, we often have to deal with geometric figures that have the “same shape” but are of different sizes. For example, an architect will make a small-scale drawing of a floor plan or a photographer will make an enlargement of a photograph. Figures that have the same shape but may be of different sizes are called similar figures . Two similar figures are illustrated in Fig. 8.20. Similar figures have corresponding angles and corresponding sides . In Fig. 8.20, triangle ABC has angles A B, , and C. Their respective corresponding angles in triangle DEF are angles D E, , and F. Sides AB BC , , and AC in triangle ABC have corresponding sides DE EF , , and DF, respectively, in triangle DEF. MATHEMATICS TODAY Computer Animation Mathematics plays a key role in the animation you see in movies such as Elio . The animated images you see on the movie screen are created using software that combines pixels (the smallest piece of a screen image) into geometric shapes, including polygons. These shapes are then stored in a computer and manipulated using various mathematical techniques so that the new shapes formed (from the original geometric shapes) approximate curves. Each movie frame has over 2 million pixels and can have over 40 million polygons. With such a huge amount of data, computers are used to carry out the mathematics needed to create animation. Why This Is Important Computer animation is just one of the many applications of polygons and other geometric concepts presented in this chapter. Triangles Acute Triangle Obtuse Triangle Right Triangle hypotenuse All angles are acute angles. One angle is an obtuse angle. One angle is a right angle. Isosceles Triangle Equilateral Triangle Scalene Triangle Two equal sides Two equal angles Three equal sides Three equal angles (60° each) No two sides are equal in length. No two angles are equal in measure. C B A F E D Figure 8.20 Definition: Similar Figures Two figures are similar if their corresponding angles have the same measure and the lengths of their corresponding sides are in proportion. In Fig. 8.20, A and D have the same measure, B and E have the same measure, and C and F have the same measure. Also, the lengths of corresponding sides of similar triangles are in proportion. When we refer to the line segment AB, we place a bar over the AB and write AB. When we refer to the length of a line segment, we do not place a bar above the two letters. For example, if we write = AB 12, we are indicating the length of line segment AB is 12. The following proportion shows that the lengths of the corresponding sides of the similar triangles in Fig. 8.20 are in proportion. = = AB DE BC EF AC DF PIXAR ANIMATION STUDIOS/Album/Alamy Stock Photo
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