Question

12
a rectangle is inscribed in a circle such that the length of the diagonal
of the rectangle is twice the length of its shortest side. the
circumference of the circle is 114π units. what is the area, in square
units, of the rectangle?
a) 57√2
b) 57√3
c) 3,249√2
d) 3,249√3

Explanation:

Step1: Find circle radius

Circumference formula: $C = 2\pi r$. Given $C = 114\pi$, so $2\pi r = 114\pi$. Divide by $\pi$: $2r = 114$, so $r = 57$. Diagonal of rectangle = diameter = $2r = 114$.

Step2: Define rectangle sides

Let shortest side = $x$, diagonal = $2x = 114$. So $x = 57$. Let longer side = $y$. By Pythagoras: $x^2 + y^2 = (2x)^2$.

Step3: Calculate longer side

Substitute $x = 57$: $57^2 + y^2 = (114)^2$. $114 = 2×57$, so $(114)^2 = 4×57^2$. Then $y^2 = 4×57^2 - 57^2 = 3×57^2$, so $y = 57√3$.

Step4: Compute area

Area = $x×y = 57×57√3 = 3249√3$.

Answer:

D. 3,249√3