Step 1: We find the area of the triangle. The given triangle is equilateral, and the formula to find the area of an equilateral triangle is:
[tex]A=\frac{\sqrt{3}}{4}a^2[/tex]Then, we have:
[tex]\begin{gathered} a=5+5=10 \\ A=\frac{3}{4}a^{2} \\ A=\frac{3}{4}(10)^2 \\ A=\frac{3}{4}(100) \\ A=\frac{300}{4} \\ A=75 \end{gathered}[/tex]Step 2: We find the area of a sector of the circle. The formula to find the area of a sector of a circle is:
[tex]\begin{gathered} \text{ Area of a sector of a circle }=\frac{\theta}{360\degree}\cdot\pi r \\ \text{ Where} \\ \theta\text{ is the angle measured in degrees} \\ \text{r is the radius of the circle} \end{gathered}[/tex]Then, we have:
[tex]\begin{gathered} \theta=60° \\ \text{ Area of a sector of circle }=\frac{\theta}{360\degree}\cdot\pi r \\ \text{ Area of a sector of circle }=\frac{60°}{360°}\cdot\pi(5) \\ \text{ Area of a sector of circle }=(\frac{1}{6})\pi(5) \\ \text{ Area of a sector of circle }=\frac{5\pi}{6} \\ \text{ or} \\ \text{ Area of a sector of circle }\approx2.62 \end{gathered}[/tex]Step 3: We find the area of the shaded region.
[tex]\begin{gathered} \text{ Area of shaded region }=\text{ Area triangle}-3\cdot\text{ Area of a sector of circle} \\ \text{ Area of shaded region }=75-3\cdot\frac{5\pi}{6} \\ \text{ Area of shaded region }\approx75-7.85 \\ \text{ Area of shaded region }\approx67.15 \end{gathered}[/tex]AnswerThe area of the shaded region is approximately 67.15 square units.
