Multiple choice

In fig 9.7, if PA and PB are tangents to the circle with centre O such that ${\angle APB = 50 ^0}$, then ${\angle OAB}$ is equal to

  1. ${25 ^0}$
  2. ${30^0}$
  3. ${40^0}$
  4. ${50 ^0}$
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A Correct answer
Explanation

In triangle OAP, angle OAP = 90 degrees. Angle APO = 50/2 = 25 degrees. In triangle OAP, angle AOP = 180 - 90 - 25 = 65 degrees. In triangle OAB, OA = OB (radii), so angle OAB = angle OBA. Angle AOB = 360 - 2 * angle AOP = 360 - 130 = 230? No, angle AOB = 180 - 50 = 130. Angle OAB = (180 - 130)/2 = 25 degrees.

AI explanation

Because PA and PB are tangents from external point P, triangle OAP is a right-angled triangle where angle OAP is 90 degrees. The line segment OA joins the center to the external point and bisects angle APB, making angle OPA equal to 25 degrees. The angles in triangle OAP must add up to 180 degrees, so angle OAP plus angle OPA plus angle AOP equals 180 degrees, yielding 90 plus 25 plus angle AOP equals 180. Because the radii form two congruent right triangles, angle OAB is the same as angle OAP, which is 25 degrees.

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