Question

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

Explanation:

Step1: Recall tangent - arc relationship

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Here, $\angle QNT$ is an angle formed by tangent $TS$ and chord $QN$ at the point of tangency $N$. Let the measure of arc $\overset{\frown}{QN}$ be $x$. Then $\angle QNT=\frac{1}{2}\overset{\frown}{QN}$.

Step2: Solve for the arc measure

Given $\angle QNT = 74^{\circ}$, and since $\angle QNT=\frac{1}{2}\overset{\frown}{QN}$, we can solve for $\overset{\frown}{QN}$ by multiplying both sides of the equation by 2. So, $\overset{\frown}{QN}=2\times\angle QNT$.
Substituting the value of $\angle QNT = 74^{\circ}$, we get $\overset{\frown}{QN}=148^{\circ}$.

Answer:

$148^{\circ}$