The shortest side of a polygon of area 196 sq. in. is 4 in. long. Find the area of a similar polygon whose shortest side is 8 in. long.
The areas of similar polygons are proportional to the square of their corresponding sides
so x/196 = 8^2/4^2
x/196 = 64/16
x = 784
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To find the area of a similar polygon, we need to understand the relationship between the sides and the areas of similar polygons.
In similar polygons, the ratio of the corresponding side lengths is equal to the square root of the ratio of their areas. Let's call the ratio of the side lengths "r" and the ratio of the areas "R".
In this case, we have two similar polygons with side lengths of 4 in. and 8 in. The ratio of the side lengths is 8/4 = 2. Therefore, the ratio of the areas is R = √(2^2) = √4 = 2.
Now, we can use this ratio to find the area of the larger polygon. We know that the area of the original polygon is 196 sq. in. Since the ratio of the areas is 2, the area of the larger polygon can be found by multiplying the area of the original polygon by the square of the ratio of the areas.
Area of larger polygon = Area of original polygon × (R^2)
Area of larger polygon = 196 × (2^2) = 196 × 4 = 784 sq. in.
Therefore, the area of the similar polygon with a shortest side of 8 in. is 784 sq. in.