Question

in circle o, ( moverarc{ab}=90^{circ} ). use the given information to determine the length of the radius of circle o. suppose the area of the segment is ( 25pi - 50 ) square inches.

Explanation:

Step1: Recall the formula for the area of a segment

The area of a segment \(A_{segment}\) of a circle with radius \(r\) and central angle \(\theta\) (in degrees) is given by \(A_{segment}=\frac{\theta}{360}\times\pi r^{2}-\frac{1}{2}r^{2}\sin\theta\). Here, \(\theta = 90^{\circ}\), so \(A_{segment}=\frac{90}{360}\times\pi r^{2}-\frac{1}{2}r^{2}\sin90^{\circ}\). Since \(\sin90^{\circ}=1\), the formula simplifies to \(A_{segment}=\frac{\pi r^{2}}{4}-\frac{r^{2}}{2}\).

Step2: Substitute the given area of the segment

We know that \(A_{segment}=25\pi - 50\). So, \(\frac{\pi r^{2}}{4}-\frac{r^{2}}{2}=25\pi - 50\). Factor out \(\frac{r^{2}}{4}\) from the left - hand side: \(\frac{r^{2}}{4}(\pi - 2)=25(\pi - 2)\).

Step3: Solve for \(r\)

Divide both sides of the equation \(\frac{r^{2}}{4}(\pi - 2)=25(\pi - 2)\) by \((\pi - 2)\) (assuming \(\pi
eq2\), which is true as \(\pi\approx3.14\)). We get \(\frac{r^{2}}{4}=25\). Then multiply both sides by 4: \(r^{2}=100\). Take the square root of both sides: \(r = 10\) (we take the positive value since \(r\) represents the radius of a circle).

Answer:

The radius of the circle \(r = 10\) inches.