6.4
Special Right Triangles
So far we have been interested in the sides of any right triangle. In this section we are going to explore the relationship among the sides of two particular right triangles. What makes these triangles special is that the angles are "nice" and we are able to compute the lengths of the sides. In addition, as we will explore, these triangles come up in a variety of problems having to do with other geometric shapes. The first special triangle is the isosceles right triangle. This triangle has two $45^\circ$ angles, and we often refer to it as the 45–45–90 triangle.
- Theorem 6.5: In a 45–45–90 triangle, the hypotenuse is $\sqrt{2}$ times as long as a leg.
Proof: Suppose we have a 45–45–90 triangle with a leg of length $x$, as shown below. Using the Pythagorean Theorem we have
\begin{align} x^2+x^2&=c^2\\ 2x^2&=c^2\\ \sqrt{2x^2}&=c\\ \sqrt{2}\cdot\sqrt{x^2}&=c\\ \sqrt{2}\cdot x&=c\\ &\text{or}\\ \sqrt{2}\cdot\text{ leg }&=\text{hypotenuse} \end{align} |
Two 45–45–90 triangles are formed whenever a diagonal is drawn in a square (and four small ones appear if both diagonals are drawn). Another place we encounter this triangle is when we extend sides of a regular octagon. Since each exterior angle of a regular octagon is a 45° angle, extending sides $\overline{AB}$ and $\overline{CD}$ of regular octagon $ABCDEFGH$ produces a 45–45–90 triangle in the exterior of the octagon.
Since the altitude bisects the side of the equilateral triangle, we let the hypotenuse of the 30–60–90 triangle be $2y$. Then the side opposite the 30° angle is $y$. Now we apply the Pythagorean Theorem to the 30–60–90 triangle and solve for $h$. That gives
\begin{align} y^2+h^2&=(2y)^2\\ y^2+h^2&=4y^2\\ h^2&=3y^2\\ h&=\sqrt{3y^2}\\ h&=\sqrt{3}\cdot y\\ &\text{or}\\ \text{ longer leg }&=\sqrt{3}\cdot\text{ shorter leg} \end{align} |
The result is the following theorem.
- Theorem 6.6: In a 30–60–90 triangle, the hypotenuse is 2 times as long as the shorter leg and the longer leg is $\sqrt{3}$ times as long as the shorter leg.
Thus we have the following two families of special right triangles:
$$30–60–90\Leftrightarrow \left(x,x\sqrt{3},2x\right)\quad\text{ and }\quad 45–45–90\Leftrightarrow \left(x, x, x\sqrt{2}\right).$$
We have noted that the 45–45–90 triangle comes up in problems with squares and octagons. As our derivation of the result makes clear, the 30–60–90 triangle is intimately related to an equilateral triangle; in addition, since a regular hexagon can be thought of as being made up of six equilateral triangles, the 30–60–90 triangle will play an important role in work with regular hexagons.
The 30–60–90 and 45–45–90 triangles are the ones that many of our exercises will depend upon. Those are the ones that have to be memorized. However, there are other "special" right triangles as we will see in some of the exercises. As mentioned at the start of this section, these are triangles for which we know the angles and can compute the sides.