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.

Answer 1

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Answer:

f(x) = (π-2)x^2/4

Explanation:

The area of the triangle is ...

A = 1/2(x)(x) = (1/2)x^2

The hypotenuse of the triangle is x√2, so the radius of the circle is x/√2. That means the area of the semicircle is ...

A = 1/2πr^2 = (π/2)(x/√2)^2 = (π/4)x^2

The shaded area is the difference between the semicircle area and the triangle area.

f(x) = (π/4)x^2 -(1/2)x^2

f(x) = (π-2)x^2/4