The shortest side of a polygon of area 196 square inches is 4 inches long. What is the area of a similar polygon whose shortest side is 8 inches long?
4 x 196 = 784 sq. in.
Areas of similar figures are proportional the square of any corresponding linear dimension.
To find the area of a similar polygon, we need to use the concept of similarity ratios. In a similar polygon, the corresponding sides are proportional.
In this case, the shortest side of the original polygon is 4 inches long, and we need to find the area of a similar polygon with a shortest side of 8 inches long.
Let's assume that the corresponding sides of the two polygons are in the ratio of x:4 (where x represents the factor by which the sides are multiplied to get the new sizes). Therefore, the corresponding sides of the two polygons are 4 inches and 8 inches respectively.
Since the area of a polygon is directly proportional to the square of its side length, the area ratio of the two polygons will be (x:4)^2.
To find the value of x, we need to solve the equation (x:4)^2 = 8^2.
Simplifying the equation, we have x^2:16 = 64.
Multiplying both sides of the equation by 16, we get x^2 = 1024.
Taking the square root of both sides, we find that x ≈ 32.
Therefore, the corresponding area ratio of the two polygons is 32:4, or simply 8:1.
Now that we know the area ratio, we can calculate the area of the new polygon. The area of the original polygon is given as 196 square inches. So, the area of the new polygon will be (area of original polygon) x (area ratio) = 196 x (8:1) = (196 x 8) / 1 = 1568 square inches.
Hence, the area of a similar polygon whose shortest side is 8 inches long is 1568 square inches.