Question

select the correct answer. what is the measure of angle qpr in the given figure? a. 32° b. 56° c. 62° d. 68°

Explanation:

Step1: Recall the tangent - secant angle theorem

The measure of an angle formed by a tangent and a secant (or a chord and a tangent) drawn from a point outside the circle is half the difference of the measures of the intercepted arcs. In this case, the angle $\angle QPR$ is formed by a tangent $PQ$ and a secant $PR$. The intercepted arcs are the major arc $QR$ (with measure $112^{\circ}$) and the minor arc $QR$. Since the total measure of a circle is $360^{\circ}$, the measure of the minor arc $QR$ is $360 - 112=248^{\circ}$? Wait, no, actually, when the angle is formed outside the circle, the formula is $\text{Angle}=\frac{1}{2}(\text{measure of the intercepted major arc}-\text{measure of the intercepted minor arc})$. Wait, no, correction: The measure of an angle formed outside the circle by a tangent and a secant is half the difference of the measures of the intercepted arcs, where the larger arc minus the smaller arc. But in this case, since $PQ$ is a tangent and $PR$ is a secant, the intercepted arcs are the arc $QR$ that is inside the angle and the rest. Wait, maybe a better way: The angle between a tangent and a chord is equal to half the measure of the intercepted arc. Wait, $PQ$ is a tangent, $QR$ is a chord. So the angle between tangent $PQ$ and chord $QR$ is equal to half the measure of the intercepted arc $QR$. But here we have angle at $P$, between secant $PR$ and tangent $PQ$. Wait, let's re - examine.

Wait, the formula for the angle formed by a tangent and a secant from an external point: If a tangent from point $P$ touches the circle at $Q$ and a secant from $P$ passes through the circle, intersecting it at $R$ and another point (but in this case, $PR$ is a secant passing through $R$ and $P$? Wait, no, $P$ is outside the circle, $PQ$ is tangent at $Q$, $PR$ is a secant intersecting the circle at $R$. Then the measure of $\angle QPR$ is half the difference of the measures of the intercepted arcs. The intercepted arcs are the arc $QR$ that is cut off by the secant and tangent. Wait, actually, the correct formula is $\angle QPR=\frac{1}{2}(\text{measure of the arc }QR\text{ (the one not containing the angle)}-\text{measure of the arc }QR\text{ (the one containing the angle)})$. Wait, no, the measure of the angle formed by a tangent and a secant drawn from a point outside the circle is equal to half the difference of the measures of the intercepted arcs. The larger arc minus the smaller arc, divided by 2.

Wait, in the diagram, the arc $QR$ that is given as $112^{\circ}$ - let's assume that the arc $QR$ (the one between $Q$ and $R$ along the circle) is $112^{\circ}$, and the other arc $QR$ (the major arc) is $360 - 112 = 248^{\circ}$. But the angle at $P$ is formed outside the circle, so $\angle QPR=\frac{1}{2}(\text{major arc }QR-\text{minor arc }QR)=\frac{1}{2}(248 - 112)=\frac{1}{2}(136) = 68$? No, that can't be. Wait, maybe I mixed up. Wait, the angle between a tangent and a secant is equal to half the measure of the intercepted arc. Wait, no, the angle between a tangent and a chord is equal to half the measure of the intercepted arc. So if $PQ$ is tangent at $Q$, and $QR$ is a chord, then the angle between $PQ$ and $QR$ is half the measure of arc $QR$. But we need angle at $P$, between $PQ$ and $PR$.

Wait, maybe the diagram is such that $PR$ is a straight line (a secant) passing through $R$ (a point on the circle) and $P$ (outside), and $PQ$ is a tangent at $Q$. Then, the angle $\angle QPR$: The sum of the angle $\angle QPR$ and the angle between $PQ$ and $QR$ (let's call it $\angle PQR$) and angle $\angle PRQ$ should be $180^{\circ}$? No, maybe a better approach.

Wait, the measure of the angle formed by a tangent and a secant from an external point is $\frac{1}{2}(\text{measure of the intercepted arc})$. Wait, no, the correct formula is: If a tangent and a secant are drawn from a point outside the circle, then the measure of the angle formed is half the difference of the measures of the intercepted arcs. The intercepted arcs are the arc that is "cut off" by the secant and tangent, with the larger arc minus the smaller arc.

In this case, the tangent is $PQ$ and the secant is $PR$. The intercepted arcs are arc $QR$ (the one between $Q$ and $R$) and the rest of the circle. Wait, let's assume that the arc $QR$ (the minor arc) is $112^{\circ}$, then the major arc $QR$ is $360 - 112=248^{\circ}$. Then the angle at $P$ is $\frac{1}{2}(\text{major arc }QR-\text{minor arc }QR)=\frac{1}{2}(248 - 112)=\frac{1}{2}(136) = 68$? No, that's not matching the options. Wait, maybe I got the formula wrong.

Wait, another approach: The angle between a tangent and a chord is equal to half the measure of the intercepted arc. So the angle between tangent $PQ$ and chord $QR$ is $\frac{1}{2}\times112 = 56^{\circ}$. Then, in triangle $PQR$? Wait, no, $PR$ is a straight line? Wait, $PR$ is a secant, so $R$ is on the circle, $P$ is outside. So $PQ$ is tangent, $PR$ is secant. Then, the angle $\angle QPR$: Let's consider that the arc $QR$ is $112^{\circ}$, the angle between tangent $PQ$ and chord $QR$ is $\frac{1}{2}\times112 = 56^{\circ}$. Then, since $PR$ is a straight line (a secant), and if we consider the angle on a straight line, but maybe not. Wait, the options are $32,56,62,68$. Wait, maybe the arc given is the major arc? Wait, no, the diagram shows the arc inside the circle as $112^{\circ}$. Wait, maybe the formula is $\angle QPR = 90^{\circ}-\frac{112^{\circ}}{2}$? No, that would be $90 - 56 = 34$, not matching. Wait, maybe the angle is formed by a tangent and a secant, and the intercepted arc is $112^{\circ}$, so the angle is $\frac{1}{2}(180 - 112)$? No, $180 - 112 = 68$, half of that is $34$. Not matching.

Wait, let's start over. The correct formula for the angle between a tangent and a secant drawn from an external point $P$ to a circle, where the tangent touches the circle at $Q$ and the secant intersects the circle at $R$ and another point (but in this case, $PR$ is a secant passing through $R$ and $P$? Wait, no, $R$ is on the circle, $P$ is outside, $PQ$ is tangent at $Q$, $PR$ is a secant passing through $R$ (so $PR$ intersects the circle at $R$ and maybe another point, but in the diagram, it looks like $PR$ is a secant passing through $R$ and $P$, with $R$ on the circle. Then the measure of $\angle QPR$ is $\frac{1}{2}(\text{measure of the arc }QR\text{ (the one not containing }P\text{)}-\text{measure of the arc }QR\text{ (the one containing }P\text{)})$. Wait, if the arc $QR$ (containing $P$) is $x$, and the arc $QR$ (not containing $P$) is $112^{\circ}$, then $x + 112=360$, so $x = 248$. Then $\angle QPR=\frac{1}{2}(248 - 112)=\frac{1}{2}(136) = 68$. But $68$ is option D. Wait, but maybe I made a mistake. Wait, no, the formula for the angle outside the circle is $\frac{1}{2}(\text{major arc}-\text{minor arc})$. So if minor arc $QR = 112^{\circ}$, major arc $QR=360 - 112 = 248^{\circ}$, then angle $=\frac{1}{2}(248 - 112)=68^{\circ}$. So that's option D.

Wait, but let's check again. The angle formed outside the circle by a tangent and a secant is half the difference of the intercepted arcs. The two intercepted arcs are the arc that is "cut off" by the secant and tangent. So the larger arc minus the smaller arc, divided by 2. So if the smaller arc is $112^{\circ}$, larger arc is $360 - 112 = 248^{\circ}$, then the angle is $\frac{248 - 112}{2}=\frac{136}{2}=68^{\circ}$. So the answer should be $68^{\circ}$, which is option D.

Step2: Calculate the angle

Using the formula for the angle formed by a tangent and a secant outside the circle: $\angle QPR=\frac{1}{2}(\text{measure of major arc }QR-\text{measure of minor arc }QR)$

Given minor arc $QR = 112^{\circ}$, major arc $QR=360 - 112 = 248^{\circ}$

$\angle QPR=\frac{1}{2}(248 - 112)=\frac{1}{2}\times136 = 68^{\circ}$

Answer:

D. $68^{\circ}$