Question

what is the measure of angle qpr in the given figure? (figure with a circle, points q, r on the circle, point p outside, angle related, options a. 32°, b. 56°, c. 62°, d. 68°)

Explanation:

Step1: Identify the exterior angle property

The angle between two tangents from an external point to a circle is supplementary to the central angle subtended by the points of contact.

Step2: Calculate angle QPR

Let the central angle ∠QOR = 112° (O is the center). Then ∠QPR = 180° - 112° = 68°? Wait, no—wait, the 112° is the angle between the two radii? Wait, no, the figure shows 112° inside the circle between Q and R, so it's the central angle. The angle between tangents PQ and PR is ∠QPR, and the formula is ∠QPR = 180° - central angle. Wait, 180-112=68? But option D is 68, but wait—wait, no, maybe the 112° is the angle between the chord QR and something? No, no, standard tangent angle: if two tangents from P touch the circle at Q and R, then ∠QPR = 180° - ∠QOR where O is center. If ∠QOR=112°, then ∠QPR=68°, which is option D. Wait, but why did I think A earlier? No, correct calculation: 180-112=68. So answer is D. 68°

Wait, correction: The angle between two tangents from an external point is equal to 180° minus the central angle subtended by the points of contact. So if central angle is 112°, then external angle is 180-112=68°. So answer is D. 68°

Step1: Recall tangent-central angle relation

Angle between two tangents = 180° - central angle.

Step2: Calculate angle QPR

∠QPR = 180° - 112° = 68°.

Answer:

A. 32°