Proportional Segments in a Right Triangle
Big Idea: When you draw an altitude from the right angle to the hypotenuse of a right triangle, you create a beautiful set of proportional relationships that connect all the segments together.
The Setup
Consider a right triangle ABC with the right angle at vertex C. Now draw the altitude (height) from C perpendicular to the hypotenuse AB, meeting it at point H.
This altitude does something remarkable — it splits the original right triangle into two smaller triangles, and all three triangles are similar to each other!
Key Definition
Altitude to the hypotenuse: In right triangle $ABC$ with $\angle C = 90°$, the segment $CH$ drawn from $C$ perpendicular to $AB$ is called the altitude to the hypotenuse.
Let's label the segments:
- $AB = c$ (the hypotenuse)
- $AC = b$ and $BC = a$ (the legs)
- $AH = p$ and $HB = q$ (projections of the legs onto the hypotenuse)
- $CH = h$ (the altitude to the hypotenuse)
Note that $p + q = c$.
Interactive visualization — available in the full lesson