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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 scalefactor?Know What?One use of scale factors and areas is scale drawings. This technique takes a small object, like thehandprint to the right, divides it up into smaller squares and then blows up the individual squares. In this KnowWhat? you are going to make a scale drawing of your own hand. Either trace your hand or stamp it on a piece ofpaper. Then, divide your hand into 9 squares, like the one to the right, probably 2in⇥2in. Take a larger piece ofpaper and blow up each square to be 6in⇥6in(meaning you need at least an 18 in square piece of paper). Onceyou have your 6in⇥6insquares drawn, use the proportions and area to draw in your enlarged handprint.583
10.3. Areas of Similar Polygonswww.ck12.orgAreas of Similar PolygonsIn Chapter 7, we learned about similar polygons. Polygons are similar when the corresponding angles are equaland the corresponding sides are in the same proportion. In that chapter we also discussed the relationship of theperimeters of similar polygons. Namely, the scale factor for the sides of two similar polygons is the same as the ratioof the perimeters.Example 1:The two rectangles below are similar. Find the scale factor and the ratio of the perimeters.Solution:The scale factor is1624, which reduces to23.The perimeter of the smaller rectangle is 52 units.Theperimeter of the larger rectangle is 78 units. The ratio of the perimeters is5278=23.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:Asmall=10·16=160units2Alarge=15·24=360units2The ratio of the areas would be160360=49.The ratio of the sides, or scale factor was23and the ratio of the areas is49. Notice that the ratio of the areas isthesquareof the scale factor. An easy way to remember this is to think about the units of area, which are alwayssquared.Therefore, you would alwayssquarethe 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 ismn, then the ratio ofthe areas would be(mn)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 findthe length of the 3rdside in the triangle and then apply the area formulas and make a ratio. The second, and easierway, would be to find the ratio of the sides and then square that.