Question

in the figure, abcd is a 6 by 16 rectangle inscribed in a semi - circle, with points a and b lying on the diameter. find the exact area of the shaded region. (figure is not to scale.) the area of the shaded region is . (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the area of the semi - circle

The diameter of the semi - circle is the length of the rectangle's diagonal. Using the Pythagorean theorem for the rectangle with length $l = 16$ and width $w = 6$, the diagonal $d$ of the rectangle (diameter of the semi - circle) is $d=\sqrt{6^{2}+16^{2}}=\sqrt{36 + 256}=\sqrt{292}=2\sqrt{73}$. So the radius of the semi - circle $r=\sqrt{73}$. The area of a semi - circle $A_{semicircle}=\frac{1}{2}\pi r^{2}=\frac{1}{2}\pi(\sqrt{73})^{2}=\frac{73\pi}{2}$.

Step2: Find the area of the rectangle

The area of the rectangle $A_{rectangle}=l\times w=16\times6 = 96$.

Step3: Find the area of the shaded region

The area of the shaded region $A = A_{semicircle}-A_{rectangle}=\frac{73\pi}{2}-96$.

Answer:

$\frac{73\pi}{2}-96$