2 Find the scale factor of the sides of the similar shapes Both figures are

2 find the scale factor of the sides of the similar

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2. Find the scale factor of the sides of the similar shapes. Both figures are squares. 3. Find the area of each square. 4. Find the ratio of the smaller square’s area to the larger square’s area. Reduce it. How does it relate to the scale factor? Know What? One use of scale factors and areas is scale drawings. This technique takes a small object, like the handprint to the right, divides it up into smaller squares and then blows up the individual squares. In this Know What? you are going to make a scale drawing of your own hand. Either trace your hand or stamp it on a piece of paper. Then, divide your hand into 9 squares, like the one to the right, probably 2 in 2 in . Take a larger piece of paper and blow up each square to be 6 in 6 in (meaning you need at least an 18 in square piece of paper). Once you have your 6 in 6 in squares drawn, use the proportions and area to draw in your enlarged handprint. 583
10.3. Areas of Similar Polygons www. c k12 .org Areas of Similar Polygons In Chapter 7, we learned about similar polygons. Polygons are similar when the corresponding angles are equal and the corresponding sides are in the same proportion. In that chapter we also discussed the relationship of the perimeters of similar polygons. Namely, the scale factor for the sides of two similar polygons is the same as the ratio of the perimeters. Example 1: The two rectangles below are similar. Find the scale factor and the ratio of the perimeters. Solution: The scale factor is 16 24 , which reduces to 2 3 . The perimeter of the smaller rectangle is 52 units. The perimeter of the larger rectangle is 78 units. The ratio of the perimeters is 52 78 = 2 3 . The ratio of the perimeters is the same as the scale factor. In fact, the ratio of any part of two similar shapes (diagonals, medians, midsegments, altitudes, etc.) is the same as the scale factor. Example 2: Find the area of each rectangle from Example 1. Then, find the ratio of the areas. Solution: A small = 10 · 16 = 160 units 2 A large = 15 · 24 = 360 units 2 The ratio of the areas would be 160 360 = 4 9 . The ratio of the sides, or scale factor was 2 3 and the ratio of the areas is 4 9 . Notice that the ratio of the areas is the square of the scale factor. An easy way to remember this is to think about the units of area, which are always squared. Therefore, you would always square the scale factor to get the ratio of the areas. Area of Similar Polygons Theorem: If the scale factor of the sides of two similar polygons is m n , then the ratio of the areas would be ( m n ) 2 . Example 2: Find the ratio of the areas of the rhombi below. The rhombi are similar. Solution: There are two ways to approach this problem. One way would be to use the Pythagorean Theorem to find the length of the 3 rd side in the triangle and then apply the area formulas and make a ratio. The second, and easier way, would be to find the ratio of the sides and then square that.

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