Question

in the diagram of circle r, m∠fgh is 50°. what is mfeh?

Explanation:

Step1: Recall the formula for the measure of an angle formed by two tangents

The measure of an angle formed by two tangents drawn from an external point to a circle is half the difference of the measures of the intercepted arcs. Let the measure of the major arc \(FEH\) be \(y\) and the measure of the minor arc \(FEH\) (the arc opposite to the angle \(\angle FGH\)) be \(x\). We know that \(y + x=360^{\circ}\) (since the sum of the measures of a major arc and its corresponding minor arc in a circle is \(360^{\circ}\)), and the formula for \(\angle FGH\) is \(m\angle FGH=\frac{1}{2}(y - x)\).

Step2: Substitute the given value of \(\angle FGH\) into the formula

Given \(m\angle FGH = 50^{\circ}\), we have \(50^{\circ}=\frac{1}{2}(y - x)\). Multiply both sides of the equation by \(2\): \(100^{\circ}=y - x\).

Step3: Solve the system of equations

We have the system of equations:
\(

$$\begin{cases}y + x=360\\y - x=100\end{cases}$$

\)
Add the two equations together: \((y + x)+(y - x)=360 + 100\).
Simplify: \(2y=460\).
Divide both sides by \(2\): \(y = 230^{\circ}\).

Answer:

\(230^{\circ}\)