Question

circle t has diameters $overline{rp}$ and $overline{qs}$. the measure of $angle rtq$ is $12^{circ}$ less than the measure of $angle rts$. what is the measure of $overset{\frown}{qp}$? 78°, 96°, 84°, 88°

Explanation:

Step1: Note the relationship between angles

Since $\angle RTQ$ and $\angle RTS$ are supplementary (because $\overline{QS}$ is a diameter and they form a linear - pair), let $\angle RTQ = x$ and $\angle RTS=y$. We know that $x + y=180^{\circ}$ and $x=y - 12^{\circ}$.

Step2: Substitute and solve for $y$

Substitute $x=y - 12^{\circ}$ into $x + y=180^{\circ}$, we get $(y - 12^{\circ})+y=180^{\circ}$.
Combining like - terms: $2y-12^{\circ}=180^{\circ}$.
Add $12^{\circ}$ to both sides: $2y=192^{\circ}$.
Divide both sides by 2: $y = 96^{\circ}$.

Step3: Recall the central - angle and arc relationship

The measure of an arc is equal to the measure of its central angle. The central angle of arc $\overparen{QP}$ is $\angle QTP$. And $\angle QTP=\angle RTS$ (vertically opposite angles). So the measure of $\overparen{QP}$ is equal to the measure of $\angle RTS$.

Answer:

$96^{\circ}$