Altitude to the hypotenuse, step by step
Drop a perpendicular from the right angle onto the hypotenuse and the triangle splits into two more, each similar to the whole and to each other.
Find the length of the altitude h.
The altitude as a geometric mean
- Dropping the altitude from the right angle onto the hypotenuse makes two smaller triangles. Each has a right angle, and each shares one of the original's acute angles — so all three triangles are similar.
- Similar means corresponding sides are in the same ratio. Writing that ratio down for the pair you want is the whole method; there is nothing to memorise beyond which pair.
- For the ALTITUDE the ratio pairs it with the two pieces: h/p = q/h, so h² = p × q.
- 576 under the root gives 24. "Geometric mean" is just the name for this: the middle term of a ratio that repeats.
How it works.
- 01
Dropping the altitude from the right angle onto the hypotenuse makes two smaller triangles. Each has a right angle, and each shares one of the original's acute angles — so all three triangles are similar.
- 02
Similar means corresponding sides are in the same ratio. Writing that ratio down for the pair you want is the whole method; there is nothing to memorise beyond which pair.
- 03
For the ALTITUDE the ratio pairs it with the two pieces: h/p = q/h, so h² = p × q.
- 04
576 under the root gives 24. "Geometric mean" is just the name for this: the middle term of a ratio that repeats.