Question

line segment ts is tangent to circle o at point n. if the measure of ∠qnt is 74°, what is the measure of arc qpn? 74° 212° 148° 37°

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a secant is half the difference of the measures of the intercepted arcs. Let the measure of arc $\widehat{QPN}$ be $x$ and assume the measure of the other intercepted arc (the minor arc $\widehat{QN}$) be $y$. The measure of $\angle QNT$ is given by $\frac{1}{2}(x - y)$. But if we consider the whole - circle property, the sum of the measures of the two arcs $x + y=360^{\circ}$, so $y = 360 - x$. And we know that $\angle QNT=74^{\circ}$.

Step2: Substitute into the formula

We have $74=\frac{1}{2}(x-(360 - x))$. First, expand the right - hand side: $74=\frac{1}{2}(x - 360+x)=\frac{1}{2}(2x - 360)=x - 180$.

Step3: Solve for $x$

Add 180 to both sides of the equation $74=x - 180$. Then $x=74 + 180=212^{\circ}$.

Answer:

$212^{\circ}$