Question

The altitude to the hypotenuse of a right triangle is the geometric mean between the segments on the hypotenuse.

A.)always
B.)sometimes
C.)never

Answer 1

If I remember correctly it is sometimes. My apologies if I'm wrong sir/ma'am

Answer 2

Option A is correct .i.e., Always

Explanation:

There is a result of similarity of triangle which states that,

If a perpendicular is drawn from the vertex of the right angle of a right triangle to the hypotenuse then triangles on both sides of the perpendicular are similar to the whole triangle and to each other.

Let ΔABC with ∠B = 90° and BD ⊥ AC

from above result,

ΔABD is similar to ΔCBD

`\implies(AD)/(BD)=(BD)/(CD)=(AB)/(CB)`

considering first equality we get,

BD² = AD . CD

This Proves that the altitude to the hypotenuse of a right triangle is always the geometric mean between the segments on the hypotenuse.

.

Therefore, Option A is correct.