Question

The figure below is made up of a circle and a triangle. The ratio of the unshaded area of the triangle to the shaded area of the triangle is 5:3. The area of the triangle is twice the area of the circle, and the total area of the figure is 126 cm². Find the area of the shaded part Understand Choose a Plan​

Answer 1

31.5 cm²

Explanation:

Let's denote the unshaded area of the triangle as 5x and the shaded area of the triangle as 3x (since the ratio of the unshaded area to the shaded area is 5:3).

Now, the area of the triangle is twice the area of the circle. Let's denote the area of the circle as A. Therefore, the area of the triangle is 2A.

The total area of the figure is the sum of the areas of the triangle and the circle. We are given that the total area is 126 cm². So we have:

2A + A = 126

Combining like terms, we get:

3A = 126

Dividing both sides by 3, we find:

A = 42

Now we know that the area of the circle is 42 cm².

To find the shaded area, we need to subtract the unshaded area from the area of the triangle:

Shaded area = 3x

Area of the triangle = 2A = 2(42) = 84

Unshaded area = Area of the triangle - Shaded area = 84 - 3x

We are given that the ratio of the unshaded area to the shaded area is 5:3:

(84 - 3x) / 3x = 5/3

Cross-multiplying, we get:

3(84 - 3x) = 5(3x)

252 - 9x = 15x

Combining like terms, we have:

24x = 252

Dividing both sides by 24, we find:

x = 10.5

Now we can find the shaded area:

Shaded area = 3x = 3(10.5) = 31.5 cm²