Geometry
Right Triangle Similarity
Big Picture
Right triangles have a lot of special characteristics, and similarity is no exception. Understanding similar right triangles is very useful in geometry and trignometry.
Key Terms
Altitude: A line segment drawn perpendicularly from a vertex of the triangle to the other side.
Geometric Mean: The nth root of the product of n numbers.
Right Triangle Altitudes
An altitude is a perpendicular segment that connects the vertex of a triangle to the opposite side. It is also known as the height of the triangle.
The altitude of right triangles has a special attribute.
Theorem: If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
Since the triangles are similar, side proportions can be used to find the lengths of any missing lengths or altitude.
\(\frac{DA}{AB} = \frac{AC}{CB} = \frac{DC}{CA}\)
\(\frac{DA}{DB} = \frac{AC}{AB} = \frac{DC}{DA}\)
\(\frac{AB}{DB} = \frac{CB}{AB} = \frac{CA}{DA}\)
Geometric Mean
The geometric mean of two positive numbers a and b is the number x so that \(\frac{a}{x} = \frac{x}{b}\)
\(\frac{a}{x} = \frac{x}{b} \implies x^2 = ab \implies x = \sqrt{ab}\)
The geometric mean can be used to FInd the altitude of a right triangle.
In a right triangle, the altitude drawn from the right angle to the hypotenuse divides the hypotenuse into two segments. In the triangle above, the hypotenuse is divided into \(\color{red}{DC}\color{black}{\text{ and }}\color{blue}{CB}\).
