What ratio signifies altitude as the mean proportional in triangle geometry?

In triangle geometry, specifically in right triangles, the altitude to the hypotenuse can be determined using the concept of the mean proportional. This scenario applies to the relationship established by similar triangles formed by the altitude, the segments of the hypotenuse, and the legs of the triangle involved.

In the context of this question, the altitude creates two smaller right triangles that are similar to the original triangle and also to each other. The correct ratio indicates that the altitude (h) serves as the mean proportional between the segments of the hypotenuse created by the foot of the altitude (let's denote them as a and b). Thus, the relationship can be expressed as:

[ \frac{\text{part of hypotenuse}}{\text{altitude}} = \frac{\text{altitude}}{\text{part hypotenuse}}. ]

This indicates that the square of the altitude is equal to the product of the segments of the hypotenuse, confirming that the altitude creates a geometric mean proportion between the two segments of the hypotenuse (noted as a and b).

This rationale directly correlates to the correct answer. The altitude is, therefore, effectively setting up a proportion where it mathematically balances the segments of