Multiple choice

PA and PB are two tangents to a circle with centre O. If ∠AOP : ∠OPB = 13 : 5, find the measure of ∠OPA.

  1. 20°

  2. 25°

  3. 45°

  4. 15°
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In triangle OAP, angle OAP = 90 degrees. Given angle AOP : angle OPB = 13:5. Since PA and PB are tangents, angle AOP = angle BOP. Thus, angle AOP : angle OPA = 13:5 is not the ratio, but angle AOP = 13x and angle OPA = 5x. In triangle OAP, 13x + 90 + 5x = 180, so 18x = 90, x = 5. Angle OPA = 5 * 5 = 25 degrees.

AI explanation

Let angle AOP be 13x and angle OPB be 5x. Because the radius is perpendicular to the tangent, triangle OAP is a right triangle at A, meaning angle OAP is 90 degrees. The angles in triangle OAP sum to 180 degrees, so 90 + 13x + angle OPA = 180, which means angle OPA = 90 - 13x. Triangles OAP and OBP are congruent, making angle OPA equal to angle OPB, so 90 - 13x = 5x, which gives 18x = 90 and x = 5. Angle OPA is 5x, which equals 25 degrees.

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