the ratio of the perimeter of two similar squares is 5 to 4. If the area of the smaller square is 32 square units, what is the area of the larger square?
The ratio of the perimeters is equal to the ratio of their sides
if the perimeters are in the ratio of 5:4, then the sides are in the ratio of 5:4
the ratio of areas of similar shapes is proportional to the square of their sides, so
x/32 = 5^2/4^2 = 25/16
x = 32(25/16) = 50
To solve this problem, we need to understand the relationship between the sides, perimeters, and areas of similar shapes.
Let's assume the side length of the smaller square is "s" units. As mentioned, the area of the smaller square is given as 32 square units. We can use this information to find the value of "s".
The area of a square is calculated by squaring the length of its side. Therefore, we have the equation:
s^2 = 32
To find the value of "s", we take the square root of both sides:
√(s^2) = √32
s = √32
Now, since we know the side length of the smaller square, we can find the perimeter of the smaller square. The perimeter of a square is given by the equation:
Perimeter = 4 * side length
So, the perimeter of the smaller square is:
Perimeter of smaller square = 4 * s
Now, let's consider the larger square. We are given that the ratio of the perimeters of the two squares is 5 to 4. This means that the perimeter of the larger square is 5/4 times the perimeter of the smaller square.
Perimeter of larger square = (5/4) * Perimeter of smaller square
Finally, let's find the area of the larger square. The area of any square is given by:
Area = side length^2
Since the side length of the larger square is related to the side length of the smaller square by the ratio of their perimeters, we can say that:
Side length of larger square = (5/4) * s
Therefore, the area of the larger square is:
Area of larger square = (Side length of larger square)^2
Substituting the value of the side length, we have:
Area of larger square = [(5/4) * s]^2
Now, we can solve for the area of the larger square by substituting the value of "s" that we found earlier:
Area of larger square = [(5/4) * √32]^2
Simplify this expression to find the area of the larger square.