6.1 The Altitude to the Hypotenuse of a Right Triangle
Right triangles play a major role in architecture and surveying. In this chapter we will explore several important properties of right triangles. The first theorem starts by taking any right triangle and drawing an altitude to its hypotenuse. The rest of this section then examines all the neat properties an altitude to the hypotenuse produces.
- Theorem 6.1:
If the altitude is drawn to the hypotenuse of a right triangle, then
the two new triangles formed are similar to the original triangle and
to each other.
Given: $\triangle
ABC$ has right angles $\angle ACB$ and
$\overline{CD}\perp\overline{AB}$. Prove: a. $\triangle ABC\sim\triangle ACD$ b. $\triangle ABC\sim\triangle CBD$ c. $\triangle ACD\sim\triangle CBD$ |
The proof of Theorem 6.1 is easier if we can "see" the triangles we're trying to show to be similar.
Not only does the altitude to the hypotenuse divide the big triangle into two smaller (and now similar) triangles; it also divides the hypotenuse into two new segments. These new segments also have a wonderful relationship. Before we examine it, let’s recall a little algebra.
Recall that two ratios, $\frac{a}{x}$ and $\frac{y}{b}$, are said to be in proportion when $\frac{a}{x}=\frac{y}{b}$. In this proportion, $x$ and $y$ are called the means, while $a$ and $b$ are the extremes. An interesting property is created when the means are equal. In the last chapter, we introduced the following notion, but since it plays a critical role in our work with right triangles, we repeat it here.
- Definition:
Let $x$, $a$, and $b$ be three positive numbers. Then $x$ is
called the geometric mean between $a$ and $b$ if and only if
$$\frac{a}{x}=\frac{x}{b}.$$
\begin{align}
x^2&=a\cdot b&&\text{ Equation 1}\\
x&=\sqrt{a\cdot b}&&\text{ Equation 2}
\end{align}
As you can see from equation (2), geometric means involve radicals and radical simplification. When simplifying, the following are standard practice.
1. Remove perfect squares from
under the radical sign.
For example: $$\sqrt{12}=\sqrt{4\cdot 3}=2\sqrt{3}$$.
For example: $$\sqrt{12}=\sqrt{4\cdot 3}=2\sqrt{3}$$.
2. Do not write answers with
fractions under the radical sign.
For example: $$\sqrt{\frac{2}{3}}=\sqrt{\frac{2\cdot 3}{3\cdot3}}=\sqrt{\frac{6}{9}}=\frac{\sqrt{6}}{3}.$$
For example: $$\sqrt{\frac{2}{3}}=\sqrt{\frac{2\cdot 3}{3\cdot3}}=\sqrt{\frac{6}{9}}=\frac{\sqrt{6}}{3}.$$
There is a third practice about which opinion is changing. In the days before calculators, one never left a fraction with a radical in the denominator. Even something as simple as $\frac{1}{\sqrt{2}}$ was taboo. The reason is that it is hard to estimate the size of the result, since dividing 1 by the square root of 2 in one's head is tough. Rather, $\frac{1}{\sqrt{2}}$ was always rewritten as $\frac{\sqrt{2}}{2}$, obtained by multiplying the numerator and denominator of the original fraction by $\sqrt{2}$. Knowing that $\sqrt{2}\approx 1.414$ makes it easy to estimate this value; it is 0.707.
- Corollary A to Theorem 6.1:
If an altitude is drawn to the hypotenuse of a right triangle, then the length of the altitude is the geometric mean between the segments of the hypotenuse.
Restatement: If $\triangle ABC$ is a right triangle and $\overline{CD}$ is the altitude drawn to the hypotenuse, then $$CD=\sqrt{AD\cdot DB}.$$ |
Plan for the proof: In mathematics, a corollary is a statement that follows readily from a previously proven statement, typically a theorem. Since this is a corollary to Theorem 6.1, we should use that theorem in our proof to establish similar triangles. These similar triangles will then allow us to set up a proportion that will lead us to the desired conclusion. The proof of this corollary is left for the exercises.
- Corollary B to Theorem 6.1:
If an altitude is drawn to the hypotenuse of a right triangle, then each leg is the geometric mean between the whole hypotenuse and the segment of the hypotenuse that is adjacent to that leg.
The proof of Corollary B, which is similar to that of Corollary A, is also left as an exercise.