Question

15 mark for review a square is inscribed in a circle. the radius of the circle is $\frac{20sqrt{2}}{2}$ inches. what is the side length, in inches, of the square? a 20 b $\frac{20sqrt{2}}{2}$ c $20sqrt{2}$ d 40

Explanation:

Step1: Find the diameter of the circle

The diameter $d$ of a circle is twice the radius $r$. Given $r = \frac{20\sqrt{2}}{2}=10\sqrt{2}$ inches, so $d = 2r=2\times10\sqrt{2}=20\sqrt{2}$ inches. When a square is inscribed in a circle, the diameter of the circle is equal to the length of the square's diagonal. Let the side - length of the square be $a$.

Step2: Use the Pythagorean theorem

For a square of side - length $a$ and diagonal $d$, by the Pythagorean theorem $d^{2}=a^{2}+a^{2}=2a^{2}$. Since $d = 20\sqrt{2}$, then $(20\sqrt{2})^{2}=2a^{2}$. Expanding, we get $800 = 2a^{2}$.

Step3: Solve for $a$

Divide both sides of the equation $800 = 2a^{2}$ by 2: $\frac{800}{2}=a^{2}$, so $a^{2}=400$. Taking the square root of both sides, $a = 20$ (we take the positive value since length cannot be negative).

Answer:

A. 20