Question

in the diagram of circle p, m∠xyz is 72°. what is the value of x? 108° 144° 216° 252°

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant of a circle is half the difference of the measures of the intercepted arcs. So, \(m\angle XYZ=\frac{1}{2}((360 - x)-x)\).

Step2: Substitute the given angle measure

We know that \(m\angle XYZ = 72^{\circ}\), so \(72=\frac{1}{2}(360 - 2x)\).

Step3: Solve the equation

First, multiply both sides of the equation by 2: \(72\times2=360 - 2x\), which gives \(144 = 360-2x\). Then, add \(2x\) to both sides: \(2x + 144=360\). Next, subtract 144 from both sides: \(2x=360 - 144=216\). Finally, divide both sides by 2: \(x = 108\).

Answer:

\(108^{\circ}\)