Multiple choice

Two circles touch each other externally at P. AB is a common tangent to the circles touching them at A and B. The value of $\displaystyle \angle APB$ is

  1. $\displaystyle { 30 }^{ \circ }$
  2. $\displaystyle { 45 }^{ \circ }$
  3. $\displaystyle { 60 }^{ \circ }$
  4. $\displaystyle { 90 }^{ \circ }$
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D Correct answer
Explanation

In two circles touching externally at P with a common tangent AB, the angle APB is always 90 degrees because the common tangent at P bisects the common tangent AB at a point that is the center of a circle passing through A, B, and P.

AI explanation

Let the radii of the two externally touching circles be R and r. The lengths of the tangents from the common external point P to the centers of the circles are R and r, and the distance between the centers is the sum of the radii, R + r. The triangle formed by the radii and the line of centers is right-angled at P because the tangent at any point on a circle is perpendicular to the radius at that point. This forms two similar right triangles, meaning angle APB equals 90 degrees.

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