Geometric Mean & the Altitude-on-Hypotenuse Relationships

The setup: three similar triangles

Start with a right triangle and drop the altitude from the right angle straight down to the hypotenuse. The altitude cuts the hypotenuse into two pieces — call them $p$ and $q$ — and it cuts the original triangle into two smaller right triangles.

Here is the key fact: the two small triangles are each similar to the original triangle, and to each other. All three triangles have the same angles, so their sides are proportional. The geometric mean relationships below are just those proportions written as equations. The figure shows the altitude $h$ dropped to the hypotenuse, splitting it into the pieces $p$ and $q$.

Rule 1: the altitude rule

The altitude is the geometric mean of the two hypotenuse pieces: $h = \sqrt{pq}$, or equivalently $h^2 = pq$. If the altitude splits the hypotenuse into pieces of length $4$ and $9$, then $h = \sqrt{4 \cdot 9} = \sqrt{36} = 6$.

This is the rule you will use most often. When a problem gives you both pieces of the hypotenuse and asks for the altitude, multiply the pieces and take the square root.

Rule 2: the leg rule

Each leg of the original triangle is the geometric mean of two lengths: the hypotenuse piece next to that leg, and the whole hypotenuse. In symbols, $\text{leg} = \sqrt{(\text{adjacent piece}) \cdot (\text{whole hypotenuse})}$.

The word adjacent is doing the work here. Each leg touches exactly one of the two hypotenuse pieces — that is the piece you pair with the whole hypotenuse. Students who grab the wrong piece get a wrong answer that still looks reasonable, so always check which piece the leg actually touches.

Worked examples

Try one yourself

Common questions