About

In Euclidean geometry, the altitude of a right triangle drawn to the hypotenuse divides the triangle into two smaller triangles that are similar to the original and to each other. This property is critical in engineering and construction for establishing perpendicular supports and analyzing load vectors.

The Geometric Mean Theorem (also known as the Altitude Rule) connects the altitude to the segments of the hypotenuse. This tool computes the altitude length (h) using the lengths of the legs (a, b) or the segments of the hypotenuse (p, q).

Formulas

The primary formula relates the altitude to the two legs and the hypotenuse. Since the area of the triangle can be calculated as both ab2 and ch2, we derive:

h = a × b√a² + b²

This ensures the altitude is perpendicular to the hypotenuse c.

Reference Data

Input Variables	Formula Used	Theorem
Legs a, b	h = abc	Area Method
Segments p, q	h = √pq	Geometric Mean
Leg a, Segment p	h = √a² − p²	Pythagorean
Hypotenuse c, Leg a	h = a √c²−a²c	Derived

Frequently Asked Questions

It is the straight line segment drawn from the right angle vertex perpendicular to the hypotenuse opposite to it. It represents the shortest distance from the right angle to the longest side.

No. This tool specifically utilizes theorems unique to right-angled triangles (Euclidean). For general triangles, you must use sine or cosine laws or Heron's formula.

No. In a right triangle, the altitude to the hypotenuse is always shorter than (or equal to, in degenerate cases) half the hypotenuse, and strictly shorter than both legs.