Question

an isosceles right triangle is inscribed in a circle. the hypotenuse of the triangle is the diameter of the circle. write a function f in terms of x that represents the area of the shaded region. leave your answer in terms of π.
f(x) = □

Explanation:

Step1: Find the area of the circle

The hypotenuse of the isosceles - right triangle is the diameter $d$ of the circle. By the Pythagorean theorem, for an isosceles right - triangle with legs of length $x$, $d^{2}=x^{2}+x^{2}=2x^{2}$, so $d = \sqrt{2}x$ and the radius $r=\frac{d}{2}=\frac{\sqrt{2}x}{2}$. The area of a circle $A_{circle}=\pi r^{2}=\pi(\frac{\sqrt{2}x}{2})^{2}=\frac{\pi x^{2}}{2}$.

Step2: Find the area of the triangle

The area of a right - triangle is $A_{triangle}=\frac{1}{2}\times base\times height$. For an isosceles right - triangle with legs of length $x$, $A_{triangle}=\frac{1}{2}x\cdot x=\frac{1}{2}x^{2}$.

Step3: Find the area of the shaded region

The area of the shaded region $f(x)$ is the area of the circle minus the area of the triangle. So $f(x)=\frac{\pi x^{2}}{2}-\frac{1}{2}x^{2}=\frac{1}{2}x^{2}(\pi - 1)$.

Answer:

$\frac{1}{2}x^{2}(\pi - 1)$