Answer:
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