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.
For similar shapes, the area ratio is proportional to the square of the length ratio for any corresponding sides.
Area(#2)/Area(#1)
= [(short side length#2/short side length#1)]^2 = (8/4)^2 = 4
Thus the larger polygon (#2) has four times that of the smaller polygon #1), or
A2 = 196 * 4 = 784 sq. in.
To find the area of a similar polygon, we can use the concept of similarity ratios.
The similarity ratio is the ratio of corresponding sides of two similar figures. In this case, since the shortest side of the original polygon is 4 in and the shortest side of the new polygon is 8 in, the similarity ratio is 8/4 = 2.
Since the sides of the polygon are directly proportional to their lengths, the ratio of their areas is the square of the ratio of their sides. Therefore, the ratio of the areas of two similar polygons is (2)^2 = 4.
Given that the area of the original polygon is 196 sq. in., we can find the area of the new polygon by multiplying the original area by the ratio of the areas.
Area of new polygon = Area of original polygon * (Similarity ratio)^2
Area of new polygon = 196 * 4
Area of new polygon = 784 sq. in.
Therefore, the area of the similar polygon with a shortest side length of 8 in. is 784 sq. in.
To find the area of a similar polygon whose shortest side is 8 inches long, we need to find the scale factor between the two polygons.
The scale factor is calculated by dividing the length of the longest side of the larger polygon by the length of the longest side of the smaller polygon.
Let's assume the longest side of the smaller polygon is 'x' inches long.
From the given information, we know that:
Area of the smaller polygon = 196 sq. in.
Shortest side of the smaller polygon = 4 inches
We can use the formula for the area of a polygon:
Area = (1/4) x sqrt[(s-a)(s-b)(s-c)(s-d)]
Where 's' is the semiperimeter of the polygon, and 'a', 'b', 'c', 'd' are the lengths of its sides.
In this case, we only have a quadrilateral since we know the shortest side and length of another side. Let's assume the other side of the quadrilateral is 'y' inches long.
Therefore, the formula becomes:
196 = (1/4) x sqrt[(4 + y)(4 + y)(y - 4)(y - 4)]
To simplify the equation, let's solve for 'y'.
Multiply both sides of the equation by 4:
784 = (4 + y)(4 + y)(y - 4)(y - 4)
Take the square root of both sides:
28 = (4 + y)(y - 4)
Expand and simplify:
28 = 4y - 16 + y^2 - 16y
0 = y^2 - 12y - 12
Solve for 'y' using the quadratic formula:
y = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = 1, b = -12, and c = -12:
y = (12 ± sqrt((-12)^2 - 4(1)(-12))) / (2(1))
y = (12 ± sqrt(144 + 48)) / 2
y = (12 ± sqrt(192)) / 2
y = (12 ± 4√3) / 2
y = 6 ± 2√3
Since 'y' represents the length of a side, it cannot be negative. Therefore, we take the positive value:
y = 6 + 2√3
We now have the length of the second side, 'y'.
To find the scale factor, we need to divide the length of the longest side of the larger polygon (8 inches) by the length of the longest side of the smaller polygon (6 + 2√3 inches):
Scale factor = (8 inches) / (6 + 2√3 inches)
To find the area of the larger polygon, multiply the scale factor by the area of the smaller polygon:
Area of the larger polygon = Scale factor × Area of the smaller polygon
So, the area of the similar polygon whose shortest side is 8 inches long can be calculated by:
Area of the larger polygon = [(8 inches) / (6 + 2√3 inches)] × 196 sq. in.