Multiple choice

In a right triangle ABC in which B = 90, a circle is drawn with AB as diameter intersecting the hypotenuse AC and P. Then the tangent to the circle at P bisects BC.

  1. True

  2. False
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A Correct answer
Explanation

In a right triangle ABC with angle B = 90, let AB be the diameter of a circle. The circle intersects AC at P. By property of circles, angle APB = 90, so BP is the altitude to the hypotenuse. The tangent at P to the circle is perpendicular to the radius OP. Geometric analysis confirms this tangent bisects BC.

AI explanation

Since angle ABC is 90 degrees, triangle ABC is right-angled at B, meaning BC is perpendicular to AB. Because AB is the diameter, angle APB is 90 degrees (angle in a semicircle), which makes AP perpendicular to the hypotenuse AC. In the right triangle ABC, the altitude BP to the hypotenuse AC makes BP squared equal to AP times CP, and the tangent at P intersects BC at its midpoint. Therefore, the tangent to the circle at P bisects BC.

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