Alex drew a circle with right triangle PRQ inscribed in it, as shown below:

Question

The figure shows a circle with points P, Q, and R on it forming an inscribed triangle. Side PQ is a chord through the center, and angle R is a right angle. Arc QR measures 80 degrees.

If the measure of arc QR is 80°, what is the measure of angle PQR? (5 points)

50°

40°

80°

70°

Answer 1

To solve for the measure of angle \( PQR \) in the inscribed right triangle \( PRQ \), we can use the properties of circles and inscribed angles.

Given that \( QR \) is the arc with a measure of \( 80° \), we can find the measure of the inscribed angle \( PQR \) that subtends this arc.

According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of the arc it subtends. Therefore, we can calculate the angle as follows:

\[ \text{Measure of angle } PQR = \frac{1}{2} \times \text{measure of arc } QR = \frac{1}{2} \times 80° = 40° \]

Thus, the measure of angle \( PQR \) is \( 40° \).

The correct answer is:

40°