line segment ts is tangent to circle o at point n. if the measure of ∠qnt is 74°, what is the measure of arc…

Explanation:

Step1: Recall tangent - secant angle formula

The measure of an angle formed by a tangent and a secant is half the measure of the intercepted arc. Let the measure of arc $\widehat{QPN}$ be $x$ and assume the other arc (the one not $\widehat{QPN}$) be $y$. The measure of $\angle QNT$ is given by $\frac{1}{2}(x - y)$. Also, $x + y=360^{\circ}$. Since $\angle QNT = 74^{\circ}$, we have $74^{\circ}=\frac{1}{2}(x - y)$.

Step2: Express $y$ in terms of $x$

From $x + y=360^{\circ}$, we get $y = 360^{\circ}-x$. Substitute $y$ into the tangent - secant formula: $74^{\circ}=\frac{1}{2}(x-(360^{\circ}-x))$.

Step3: Solve the equation

First, expand the right - hand side: $74^{\circ}=\frac{1}{2}(x - 360^{\circ}+x)=\frac{1}{2}(2x - 360^{\circ})=x - 180^{\circ}$. Then, add $180^{\circ}$ to both sides of the equation: $x=74^{\circ}+180^{\circ}=254^{\circ}$. But we made a wrong start. The correct formula for the angle formed by a tangent and a secant is that the measure of the angle between a tangent and a secant is half of the difference of the measures of the intercepted arcs. The measure of $\angle QNT$ is half of the measure of the larger arc minus the smaller arc. Here, $\angle QNT$ is an angle formed by tangent $TS$ and secant $QN$. The measure of $\angle QNT$ is half of the measure of arc $\widehat{QPN}$. So if $\angle QNT = 74^{\circ}$, then the measure of arc $\widehat{QPN}=2\times74^{\circ}=148^{\circ}$.

Answer:

$148^{\circ}$