Question

Let $ABC$ be a right-angled triangle with hypotenuse $BC$ of length $20	ext{ cm}$. If $AP$ is perpendicular on $BC$, then the maximum possible length of $AP$, in cm, is

Accepted Answer

A. Any right-angled triangle $ABC$ with hypotenuse $BC$ can be inscribed in a circle with $BC$ as diameter (radius $R = 10	ext{ cm}$). $A$ lies on the circumference. The perpendicular distance $AP$ from $A$ to $BC$ is maximized when $A$ is at the top of the semicircle, where $AP = R = 10	ext{ cm}$.

Maximum altitude of right-angled triangle

Explanation

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