Question

Angle ABC is similar to angle DEF. The perimeter of angle ABC is five times the perimeter of angle DEF. The area of angle ABC is 100 square centimeters. What is the area of angle DEF?

Answer 1

If the area of angle ABC is 100 square centimeters and it is similar to angle DEF with a perimeter five times smaller, the area of angle DEF is 4 square centimeters.

Step-by-step explanation:

When two angles are similar, their corresponding sides are in proportion, and the ratio of their perimeters is equal to the ratio of their corresponding sides. Here, the perimeter of angle ABC is five times the perimeter of angle DEF, given that ABC is similar to DEF.

Let's assume the scale factor between the two similar angles is k. Therefore, the ratio of their perimeters is k for ABC and 1 for DEF. Since the perimeter of ABC is five times the perimeter of DEF, we get the equation k * Perimeter_DEF = Perimeter_ABC = 5 * Perimeter_DEF. Solving for k gives us k = 5.

Now, for similar figures, the ratio of their areas is the square of the scale factor. Given that the area of angle ABC is 100 square centimeters, and the scale factor is 5, the ratio of the areas is 5^2 = 25. Therefore, the area of angle DEF is 1/25 times the area of ABC, which is 100 square centimeters, resulting in an area of 4 square centimeters for angle DEF.