Question

sally drew a circle with right triangle prq inscribed in it, as shown below.

if the measure of arc qr is \\(40^\circ\\), what is the measure of angle pqr? (5 points)

\\(20^\circ\\)
\\(40^\circ\\)
\\(50^\circ\\)
\\(70^\circ\\)

Explanation:

Identify the properties of the inscribed right triangle

Using the Circle Geometry knowledge point

$$ \angle PRQ = 90^\circ $$

Since \(\triangle PRQ\) is a right triangle inscribed in the circle with the right angle at \(R\), the hypotenuse \(PQ\) must be a diameter of the circle. Therefore, the arc \(PRQ\) is a semicircle, which means:

$$ m\widehat{PRQ} = 180^\circ $$

Calculate the measure of inscribed angle QPR

Using the Arc Measure Calculations knowledge point

$$ m\widehat{QR} = 40^\circ $$

The angle \(\angle QPR\) is an inscribed angle that intercepts the arc \(\widehat{QR}\). By the Inscribed Angle Theorem:

$$ m\angle QPR = \frac{1}{2} \cdot m\widehat{QR} = \frac{1}{2} \cdot 40^\circ = 20^\circ $$

Solve for the measure of angle PQR

In the right triangle \(\triangle PRQ\), the sum of the acute angles is \(90^\circ\):

$$ m\angle PQR + m\angle QPR = 90^\circ $$

$$ m\angle PQR + 20^\circ = 90^\circ $$

$$ m\angle PQR = 70^\circ $$

Answer:

• (A) \(20^\circ\)
• (B) \(40^\circ\)
• (C) \(50^\circ\)
• (D) \(70^\circ\) (Correct answer)