Multiple choice

Write True or False and justify your answer in each of the following : If angle between two tangents drawn from a point P to a circle of radius a and centre O is $90^{0}$, then OP$= a\sqrt{2}$.

  1. True

  2. False

  3. Ambiguous

  4. Data insufficient
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A Correct answer
Explanation

The tangents from an external point P to a circle are equal in length. The radius to the point of tangency is perpendicular to the tangent. This forms a square with the radius and the tangents, where the distance OP is the diagonal of a square with side a, so OP = a*sqrt(2).

AI explanation

Since the two tangents drawn from point P are perpendicular to the respective radii at the points of contact, a square is formed by the radii, the tangent segments, and the line OP. In this quadrilateral, all interior angles are 90 degrees and the two adjacent sides from the center to the points of contact are both length a. Because this shape is a square, the diagonal OP is equal to a times the square root of 2, making the statement true.

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