Multiple choice general knowledge math & puzzles

A line segment that is tangent to a circle O at its midpoint has its endpoints at A and B. If AB = the diameter of circle O, then triangle OAB is a right triangle.

  1. True

  2. False
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A Correct answer
Explanation

The radius drawn to a point of tangency is perpendicular to the tangent line. Since the tangent is perpendicular to radius at the midpoint, and AB is a line segment, angle OAB is 90°, making triangle OAB a right triangle.

AI explanation

A radius drawn to the point of tangency is always perpendicular to the tangent line, so OT (from center O to the tangency point T, the midpoint of AB) is perpendicular to AB. Since AB equals the diameter, AT = TB = radius, and OT is also a radius, so OA and OB both equal radius times the square root of 2. Checking OA² + OB² against AB² shows they are equal, which by the converse of the Pythagorean theorem means the angle at O is a right angle, making triangle OAB a right triangle.

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