Question

in circle p, the measure of ∠qps is 15° and rq is a diameter. find the arc measures. mqt = ° mrt = ° mqtr = °

Explanation:

Step1: Recall central - angle and arc - measure relationship

The measure of an arc is equal to the measure of its central angle.

Step2: Find \(m\widehat{QT}\)

Since \(\angle QPT = 105^{\circ}\), then \(m\widehat{QT}=105^{\circ}\) (by the central - angle and arc - measure relationship).

Step3: Find \(m\widehat{RT}\)

\(\angle QPS = 15^{\circ}\), and \(\angle RPS=180^{\circ}-\angle QPS = 180 - 15=165^{\circ}\). Also, \(\angle RPT=\angle RPS-\angle SPT\). Since \(\angle QPT = 105^{\circ}\) and \(\angle QPS = 15^{\circ}\), then \(\angle SPT=\angle QPT-\angle QPS=105 - 15 = 90^{\circ}\). So \(m\widehat{RT}=180^{\circ}- 105^{\circ}=75^{\circ}\).

Step4: Find \(m\widehat{QTR}\)

The measure of a full - circle is \(360^{\circ}\). Since \(m\widehat{QT}=105^{\circ}\) and \(m\widehat{TR}=255^{\circ}\) (because \(m\widehat{QTR}=360^{\circ}-m\widehat{QT}\)), \(m\widehat{QTR}=255^{\circ}\).

Answer:

\(m\widehat{QT}=105^{\circ}\), \(m\widehat{RT}=75^{\circ}\), \(m\widehat{QTR}=255^{\circ}\)