Question

right triangle nop is inscribed in circle r. find the area of the shaded region. round your answer to the nearest tenth if necessary.

Explanation:

Step1: Find the hypotenuse (diameter of the circle)

By the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 6\), \(b=8\).
\(c=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10\). So the diameter \(d = 10\), and the radius \(r=\frac{d}{2}=5\).

Step2: Calculate the area of the circle

The area of a circle is \(A_{circle}=\pi r^{2}\). Substituting \(r = 5\), we get \(A_{circle}=\pi\times5^{2}=25\pi\approx25\times3.14 = 78.5\).

Step3: Calculate the area of the right - triangle

The area of a right - triangle is \(A_{triangle}=\frac{1}{2}\times base\times height\). Here, base \(= 6\) and height \(= 8\), so \(A_{triangle}=\frac{1}{2}\times6\times8=24\).

Step4: Calculate the area of the shaded region

The area of the shaded region \(A=A_{circle}-A_{triangle}\).
\(A = 78.5-24=54.5\).

Answer:

\(54.5\)