What is the length of minor arc AB?

Question

Circle O has a radius of 9. From an external point P, segment PA is tangent to the circle at point A and segment PB is tangent to the circle at point B. The measure of angle APB is 40°.

Accepted Answer

C. 7π. A tangent is perpendicular to the radius at the point of tangency, so angle OAP = angle OBP = 90°. In quadrilateral OAPB the four angles sum to 360°, so the central angle AOB = 360° − 90° − 90° − 40° = 140°. Minor arc AB = (140/360)(2π · 9) = (7/18)(18π) = 7π. (A) 2π = (40/360)(18π) — the student used the 40° angle at P as if it were the central angle. (B) 3.5π = (70/360)(18π) — the student halved 140°, misapplying the inscribed-angle theorem to a central angle. (D) 11π = (220/360)(18π) is the major arc AB, not the minor arc.

Answer

A. 2π — You treated the 40° angle at P as the central angle. The angle at the center, AOB, is 360° − 90° − 90° − 40° = 140°, since both radii meet the tangents at right angles.

Answer

B. 3.5π — You halved the 140° central angle, which is the inscribed-angle relationship — but AOB is already a central angle, so the arc equals 140° directly.

Answer

D. 11π — You used the reflex angle of 220°, which gives the major arc. The minor arc corresponds to the 140° central angle.

Geometry and Trigonometry practice question

Answer: C. 7π

Why the other answers are wrong

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