Question

a rectangle is inscribed in a circle, such that each vertex of the rectangle lies on the circumference of the circle. the diagonal of the rectangle is twice the length of the shortest side of the rectangle. the area of the rectangle is 1,089\sqrt{3} square units. what is the length, in units, of the diameter of the circle?

Explanation:

Step1: Let the short - side of rectangle be $a$.

The diagonal $d = 2a$.

Step2: Use Pythagorean theorem.

Let the long - side be $b$. Then $a^{2}+b^{2}=d^{2}$. Since $d = 2a$, we have $a^{2}+b^{2}=(2a)^{2}$, so $b^{2}=4a^{2}-a^{2}=3a^{2}$, and $b = \sqrt{3}a$.

Step3: Calculate the area of the rectangle.

The area of the rectangle $A=ab$. Substituting $b = \sqrt{3}a$, we get $A=a\times\sqrt{3}a=\sqrt{3}a^{2}$. Given $A = 1089\sqrt{3}$, then $\sqrt{3}a^{2}=1089\sqrt{3}$. Canceling out $\sqrt{3}$ on both sides, we have $a^{2}=1089$, so $a = 33$.

Step4: Find the diameter of the circle.

The diameter of the circle is equal to the diagonal of the rectangle. Since $d = 2a$ and $a = 33$, the diameter $d=66$.

Answer:

66