Question

rectangle jklm is inscribed in circle n. find the area of the shaded region. round your answer to the nearest tenth if necessary. answer attempt 1 out of 2 units² submit answer

Explanation:

Step1: Find the diameter of the circle

The diagonal of the rectangle is the diameter of the circle. Using the Pythagorean theorem \(d=\sqrt{32^{2}+24^{2}}\), where \(d\) is the diagonal. So \(d = \sqrt{1024 + 576}=\sqrt{1600}=40\), and the radius \(r=\frac{d}{2}=20\).

Step2: Calculate the area of the circle

The area of a circle formula is \(A_{circle}=\pi r^{2}\). Substituting \(r = 20\), we get \(A_{circle}=\pi\times20^{2}=400\pi\).

Step3: Calculate the area of the rectangle

The area of a rectangle formula is \(A_{rectangle}=l\times w\). Here \(l = 32\) and \(w = 24\), so \(A_{rectangle}=32\times24 = 768\).

Step4: Calculate the area of the shaded region

The area of the shaded region \(A = A_{circle}-A_{rectangle}\). So \(A=400\pi - 768\). Using \(\pi\approx3.14\), we have \(A\approx400\times3.14-768=1256 - 768 = 488\).

Answer:

\(488.0\)