Geometry / Unit 9: Right Triangles and Trigonometry
9.4 Similar Right Triangles
Notes
The altitude to the hypotenuse of a right triangle forms two smaller triangles. All three triangles are similar.
- Altitude rule: the altitude is the geometric mean of the two hypotenuse segments: $\displaystyle h = \sqrt{xy}$.
- Leg rule: each leg is the geometric mean of the whole hypotenuse and the segment next to that leg: $\displaystyle a = \sqrt{c \cdot x}$.
- Example: segments 4 and 9 give altitude $\displaystyle \sqrt{36} = 6$.
Practice
The altitude to the hypotenuse divides the hypotenuse into segments of 3 and 12. Find the altitude.
$\displaystyle \sqrt{3 \cdot 12} = \sqrt{36} = 6$.
The altitude to the hypotenuse divides it into segments of 4 and 12. Find the leg next to the segment of 4.
The hypotenuse is 16, so the leg is $\displaystyle \sqrt{4 \cdot 16} = 8$.
Why are the three triangles formed by the altitude to the hypotenuse similar?
Each small triangle shares an acute angle with the large one and has a right angle.
Ready for more? Take the Unit 9 test.