Question

question 5 of 5
drag each tile to the correct box.
order the measure of the three arcs from greatest to least.
178°
m(pr)
m(qr)
m(pq)

Explanation:

Step1: Recall inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. In \(\triangle PQR\) inscribed in the circle, \(\angle QPR = 34^{\circ}\), and the arc \(\overset{\frown}{QR}\) is intercepted by \(\angle QPR\), so \(m(\overset{\frown}{QR})=2\times\angle QPR = 68^{\circ}\).

Step2: Find the measure of major - arc \(\overset{\frown}{PR}\)

The given arc \(178^{\circ}\) is the major - arc \(\overset{\frown}{PR}\).

Step3: Find the measure of \(\overset{\frown}{PQ}\)

The sum of the measures of the arcs of a circle is \(360^{\circ}\). Let \(m(\overset{\frown}{PQ})=x\). We know \(m(\overset{\frown}{PR}) = 178^{\circ}\) and \(m(\overset{\frown}{QR})=68^{\circ}\). Then \(x + 178^{\circ}+68^{\circ}=360^{\circ}\), so \(x=m(\overset{\frown}{PQ})=360^{\circ}-(178^{\circ} + 68^{\circ})=114^{\circ}\).

Step4: Order the arcs

Comparing the measures \(178^{\circ}\) (for \(m(\overset{\frown}{PR})\)), \(114^{\circ}\) (for \(m(\overset{\frown}{PQ})\)) and \(68^{\circ}\) (for \(m(\overset{\frown}{QR})\)), we have \(m(\overset{\frown}{PR})>m(\overset{\frown}{PQ})>m(\overset{\frown}{QR})\).

Answer:

\(m(\overset{\frown}{PR})>m(\overset{\frown}{PQ})>m(\overset{\frown}{QR})\)