Area of a Composite Figure – Larger Triangle and Two Semicircles

424.73 yd²

Explanation

The shaded figure consists of a right triangle and two semicircular regions:

\[A_{\text{total}} = A_{\text{triangle}} + A_{\text{smaller semicircle}} + A_{\text{larger semicircle}}\]

Area of the triangle (legs 10 yd and 24 yd — note: \(10^2 + 24^2 = 676 = 26^2\), confirming it is a right triangle):

\[A_{\text{triangle}} = \frac{1}{2}(10)(24) = \frac{1}{2}(240) = 120 \text{ yd}^2\]

Area of the smaller semicircle (diameter = 10 yd, so radius = 5 yd):

\[A_{\text{smaller}} = \frac{1}{2}\pi(5)^2 = \frac{1}{2}\pi(25) = 12.5\pi\]

Area of the larger semicircle (diameter = 26 yd, so radius = 13 yd):

\[A_{\text{larger}} = \frac{1}{2}\pi(13)^2 = \frac{1}{2}\pi(169) = 84.5\pi\]

Total area:

\[A_{\text{total}} = 120 + 12.5\pi + 84.5\pi = 120 + 97\pi \approx 424.7300\]

To the nearest hundredth, the area of the shaded figure is 424.73 yd².