The Proof of the Right Triangle Altitude Theorem

The geometric mean of two positive integers $latex a$ and $latex b$ is $latex \sqrt{ab}$. 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,

$latex h = \sqrt{xy}$, $latex a = \sqrt{cx}$ and $latex b = \sqrt{cy}$.

Proof

Let $latex a$ and $latex b$ be the legs of the triangle $latex ABC$ with hypotenuse $latex c$. Let $latex h$ be the altitude of the triangle to the hypotenuse dividing the $latex c$ to segments $latex x$ and $latex y$.

right triangle altitude theorem

First, we show that that the three triangles, $latex \triangle ABC$, $latex \triangle ACD$, and $latex \triangle CBD$ are similar triangles.

$latex m \angle A = m \angle BCD$ since both of them equal $latex 90 – m \angle B$. So,  $latex \triangle ABC$, $latex \triangle ACD$, and $latex \triangle CBD$ are right triangles which all include  an angle whose measure equal $latex m \angle A$. Therefore, these three triangles are similar. Since these three triangles are similar, their sides are proportional.

It follows that for $latex \triangle ACD$ and $latex \triangle CBD$,

$latex \displaystyle \frac{x}{h} =\frac{h}{y}$

which implies that

$latex h^2 = xy$ and $latex h = \sqrt{xy}$.

It remains to show that $latex a = \sqrt{cx}$ and $latex b = \sqrt{cy}$.

For $latex \triangle ABC$ and $latex \triangle CBD$, we have

$latex \displaystyle \frac{x}{a} = \frac{a}{c}$

which means that

$latex a^2 = xc$ and $latex a = \sqrt{xc}$.

Also, for $latex \triangle ABC$ and $latex \triangle ACD$, we have

$latex \displaystyle \frac{b}{c} = \frac{y}{b}$

which means that $latex b^2 = cy$ and that $latex b = \sqrt{cy}$.

These prove the Right Triangle Altitude Theorem..

 

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