Two similar triangles have perimeters of 60 and 36 inches. Find the area of the smaller triangle if the corresponding area of the larger triangle is 100 sq. inches.
a. 90 sq. in.
b. 48 sq. in.
c. 36 sq. in.
d. 60 sq. in.
To find the area of the smaller triangle, we need to use the concept of similarity between triangles. When two triangles are similar, their corresponding sides are proportional.
Let's assume that the larger triangle has sides a, b, and c, and the smaller triangle has sides x, y, and z. Then we can set up the following proportion:
x/a = y/b = z/c
We are given that the perimeters of the larger and smaller triangles are 60 and 36 inches, respectively. The perimeters are the sums of the lengths of all the sides, so we can write the following equations:
a + b + c = 60
x + y + z = 36
To solve for the sides of the smaller triangle, we need to find the ratio between the perimeters:
( x + y + z ) / ( a + b + c ) = 36 / 60 = 3 / 5
Now we can set up the following proportional equations:
x / a = y / b = z / c = 3 / 5
We know that the area of a triangle is given by the formula A = 1/2 * base * height. Since the base and height are proportional, the area will be proportional to the square of the ratio between the sides.
Let's assume that the area of the larger triangle is A_larger, and the area of the smaller triangle is A_smaller. We are given that A_larger = 100 sq. inches.
We can set up the following proportion to find the ratio between the areas:
A_smaller / A_larger = (x^2 / a^2) = (y^2 / b^2) = (z^2 / c^2)
A_smaller / 100 = (3/5)^2
A_smaller = 100 * (3/5)^2
A_smaller = 100 * 9/25
A_smaller = 36 sq. inches
Therefore, the area of the smaller triangle is 36 sq. inches, which corresponds to option c.