Question

in the diagram, $\triangle pqr$ is inscribed in a circle. the ratio of the angle measures of $\triangle pqr$ is $4:7:7$, the measure of $\overset\frown{qr}$ is \boxed{} degrees.

Explanation:

Step1: Find the measures of the angles of \(\triangle PQR\)

Let the angles of \(\triangle PQR\) be \(4x\), \(7x\), and \(7x\).
Since the sum of angles in a triangle is \(180^{\circ}\), we have \(4x + 7x+7x=180^{\circ}\).
Combining like terms gives \(18x = 180^{\circ}\).
Solving for \(x\), we get \(x=\frac{180^{\circ}}{18}=10^{\circ}\).
So, \(\angle P = 4x=40^{\circ}\), \(\angle Q=\angle R = 7x = 70^{\circ}\).

Step2: Use the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc.
The inscribed angle \(\angle P\) intercepts arc \(QR\).
By the inscribed - angle theorem, if \(\angle P\) is an inscribed angle and \(\overset{\frown}{QR}\) is the intercepted arc, then \(\angle P=\frac{1}{2}\overset{\frown}{QR}\).

Step3: Calculate the measure of arc \(QR\)

Since \(\angle P = 40^{\circ}\) and \(\angle P=\frac{1}{2}\overset{\frown}{QR}\), then \(\overset{\frown}{QR}=2\angle P\).
Substituting \(\angle P = 40^{\circ}\), we get \(\overset{\frown}{QR}=80^{\circ}\).

Answer:

\(80\)