Question

a triangle is placed in a semicircle with a radius of 7 yd, as shown below. find the area of the shaded region. use the value 3.14 for π, and do not round your answer. be sure to include the correct unit in your answer.

Explanation:

Step1: Calculate area of semicircle

The formula for the area of a full circle is \( A = \pi r^2 \), so for a semicircle, it's \( \frac{1}{2}\pi r^2 \). Given \( r = 7 \) yd and \( \pi = 3.14 \), we substitute:
\( \frac{1}{2} \times 3.14 \times 7^2 = \frac{1}{2} \times 3.14 \times 49 = 76.93 \) \( \text{yd}^2 \)

Step2: Calculate area of triangle

The triangle has a base equal to the diameter of the semicircle (\( 2r = 14 \) yd) and height equal to the radius (\( 7 \) yd). The area of a triangle is \( \frac{1}{2} \times \text{base} \times \text{height} \):
\( \frac{1}{2} \times 14 \times 7 = 49 \) \( \text{yd}^2 \)

Step3: Subtract triangle area from semicircle area

The shaded area is semicircle area minus triangle area:
\( 76.93 - 49 = 27.93 \) \( \text{yd}^2 \)

Answer:

\( 27.93 \) square yards (or \( 27.93 \text{ yd}^2 \))