The geometric mean of two positive integers and is . In this post, we are going to show the relationship between geometric mean and the relationships among the sides and altitude of a right triangle.
The Right Triangle Altitude Theorem
In a right triangle,
- the altitude to the hypotenuse is the geometric mean of the segments into which it divides (the hypotenuse); and
- each leg is the geometric mean of the hypotenuse and the segment of the hypotenuse adjacent to the leg.
The two theorems above, states that in the triangle below,
, and .
Let and be the legs of the triangle with hypotenuse . Let be the altitude of the triangle to the hypotenuse dividing the to segments and .
First, we show that that the three triangles, , , and are similar triangles.
since both of them equal . So, , , and are right triangles which all include an angle whose measure equal . Therefore, these three triangles are similar. Since these three triangles are similar, their sides are proportional.
It follows that for and ,
which implies that
It remains to show that and .
For and , we have
which means that
Also, for and , we have
which means that and that .
These prove the Right Triangle Altitude Theorem..