Question

a rectangle is inscribed in a circle. the length of the shortest side of the rectangle is 48, and the length of the longest side of the rectangle is 48√3. what is the diameter of the circle?

Explanation:

Step1: Apply the Pythagorean theorem

When a rectangle is inscribed in a circle, the diagonal of the rectangle is equal to the diameter of the circle. Let \(a = 48\) and \(b=48\sqrt{3}\) be the sides of the rectangle. By the Pythagorean theorem \(d^{2}=a^{2}+b^{2}\), where \(d\) is the diagonal (diameter of the circle).
Substitute \(a = 48\) and \(b = 48\sqrt{3}\) into the formula:

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Step2: Solve for \(d\)

Take the square - root of both sides: \(d=\sqrt{48^{2}\times4}\). Since \(\sqrt{48^{2}\times4}=\sqrt{48^{2}}\times\sqrt{4}\), and \(\sqrt{48^{2}} = 48\), \(\sqrt{4}=2\), then \(d=48\times2=96\).

Answer:

\(96\)