If the circumference of a circle is 10π, what is the area of the largest square that can be inscribed in the circle?
Correct: B
First, find the radius of the circle from its circumference.
C = 2πr
10π = 2πr
r = 5
When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle.
Diameter d = 2r = 2 * 5 = 10.
Let 's' be the side length of the square. By the Pythagorean theorem, the diagonal of a square is s*sqrt(2).
So, s*sqrt(2) = 10
s = 10 / sqrt(2)
s = (10*sqrt(2)) / 2
s = 5*sqrt(2)
The area of the square is A = s^2.
A = (5*sqrt(2))^2
A = 5^2 * (sqrt(2))^2
A = 25 * 2
A = 50 square units.