in circle p, the measure of $angle qps$ is $15^circ$ and $overline{rq}$ is a diameter.\nfind the arc…

in circle p, the measure of $angle qps$ is $15^circ$ and $overline{rq}$ is a diameter.\nfind the arc measures.\n$mwidehat{qt} = square^circ$\n$mwidehat{rt} = square^circ$\n$mwidehat{qtr} = square^circ$

in circle p, the measure of $angle qps$ is $15^circ$ and $overline{rq}$ is a diameter.\nfind the arc measures.\n$mwidehat{qt} = square^circ$\n$mwidehat{rt} = square^circ$\n$mwidehat{qtr} = square^circ$

Answer

Explanation:

Step1: Use diameter's straight angle

Since $\overline{RQ}$ is a diameter, $m\angle RPQ = 180^\circ$.

Step2: Calculate $\angle QPT$

$\angle QPT = 180^\circ - m\angle RPT = 180^\circ - 105^\circ = 75^\circ$ Central angle = arc measure, so $m\widehat{QT} = 75^\circ$.

Step3: Identify arc $\widehat{RT}$ measure

Central angle $\angle RPT = 105^\circ$, so $m\widehat{RT} = 105^\circ$.

Step4: Calculate major arc $\widehat{QTR}$

Total circle is $360^\circ$, so $m\widehat{QTR} = 360^\circ - m\widehat{QT} = 360^\circ - 15^\circ = 345^\circ$. *(Note: $m\angle QPS=15^\circ$ means $m\widehat{QS}=15^\circ$, so the minor arc opposite $\widehat{QTR}$ is $\widehat{QS}$? No, correction: $\widehat{QTR}$ is major arc for $\widehat{QS}$? No, correction: $\widehat{QTR}$ is the major arc from Q to R through T, so it is $360^\circ - m\widehat{QS} = 360^\circ -15^\circ=345^\circ$)

Answer:

$m\widehat{QT} = 75^\circ$ $m\widehat{RT} = 105^\circ$ $m\widehat{QTR} = 345^\circ$