3. In \( \triangle X Y Z \), if \( \frac{2 X}{\sin X}=8 \mathrm{~cm} \), then the area of its circumcircle equals \( \begin{array}{llll}\text { (a) } 16 \pi & \text { (b) } 8 \pi & \text { (c) } 4 \pi & \text { (d) } 64 \pi\end{array} \)

Given that in triangle XYZ, \( \frac{2X}{\sin X} = 8 \) cm, we need to find the area of the circumcircle of the triangle. First, let's find the value of \( X \) using the given equation: \[ \frac{2X}{\sin X} = 8 \] To solve for \( X \), we can rearrange the equation: \[ 2X = 8 \sin X \] Now, we can solve for \( X \) by dividing both sides by 2: \[ X = 4 \sin X \] To find the area of the circumcircle, we need the radius of the circumcircle. The radius of the circumcircle is given by \( R = \frac{a}{2 \sin A} \), where \( a \) is the side opposite angle \( A \). In this case, the side opposite angle \( X \) is \( X \) itself. Therefore, the radius of the circumcircle is: \[ R = \frac{X}{2 \sin X} \] Now, we can find the area of the circumcircle using the formula \( A = \pi R^2 \): \[ A = \pi \left( \frac{X}{2 \sin X} \right)^2 \] Substitute the value of \( X \) into the formula to find the area of the circumcircle. Simplify the expression by following steps: - step0: Solution: \(\pi \left(\frac{4\sin\left(X\right)}{2\sin\left(X\right)}\right)^{2}\) - step1: Reduce the fraction: \(\pi \times 2^{2}\) - step2: Evaluate the power: \(\pi \times 4\) - step3: Multiply: \(4\pi \) The area of the circumcircle is \( 4\pi \) cm². Therefore, the correct answer is (c) \( 4\pi \).