Question

The altitude drawn to the hypotenuse of a right triangle divides the hypotenuse into segments of lengths 4 and 12. Find the length of the shorter leg.

Answer 1

The altitude will be the geometric mean of the lengths, so
$h=\sqrt{4 \cdot 12}=4\sqrt{3}$.  The smaller leg is the hypotenuse of a right triangle sides 4 and $4 \sqrt{3},$ so the usual trig cliche, hypotenuse 8.

Answer: 8