Question

a chord with length 8 centimeters is 3 centimeters from the center of the circle. find the area of the circle in square centimeters.

Explanation:

Step1: Use the property of chord - distance from the center

The perpendicular from the center of a circle to a chord bisects the chord. So, half - length of the chord \(l=\frac{8}{2} = 4\) cm and the distance from the center to the chord \(d = 3\) cm.

Step2: Apply the Pythagorean theorem

Let the radius of the circle be \(r\). By the Pythagorean theorem \(r^{2}=l^{2}+d^{2}\). Substitute \(l = 4\) and \(d = 3\) into the formula: \(r^{2}=4^{2}+3^{2}=16 + 9=25\).

Step3: Calculate the area of the circle

The area of a circle is \(A=\pi r^{2}\). Since \(r^{2}=25\), then \(A = 25\pi\) square centimeters.

Answer:

\(25\pi\) (i.e., the third option \(25\pi\))