8th Grade - Special Right Triangles

Introduction

Aside from the Pythagorean Theorem, there are helpful theorems we can use to solve for the measure of side lengths or angles of a right triangle.
Two special triangles are important in geometry - $30 ° - 60 ° - 90 °$ Triangle, and $45 ° - 45 ° - 90 °$ Triangle.
$45 ° - 45 ° - 90 °$ Triangle is also called the Isosceles Right Triangle.

$30 ° - 60 ° - 90 °$ Triangle Theorem - In $30 ° - 60 ° - 90 °$ Triangle, the side opposite the $30 °$ angle is half as long as the hypotenuse, and the side opposite the $60 °$ angle is $\sqrt{3}$ times as long as the side opposite the $30 °$ angle.

The converse of the $30 ° - 60 ° - 90 °$ Triangle - If one leg is half as long as the hypotenuse in a right triangle, then the opposite angle has a measure of 30 degrees.
Isosceles Right Triangle Theorem - In an isosceles right triangle, the hypotenuse is $\sqrt{2}$ times as long as either of the legs. To illustrate, we have $A C = B C$ with hypotenuse $A B = x \sqrt{2}$ .

Example 1. Refer to the figure above. If $c = 30$ , what is the length of $a$ and $b$ ?

Solution:

Since the length of the hypotenuse is $30$ units, the length of $a$ is $15$ units.

If $a = 15$ units, then $b = 15 \sqrt{3}$ units.

Example 2. Refer to the figure above. If the length of leg $b$ is 24 units, what are the lengths of the hypotenuse and the other leg?

Solution:

$△ A C B$ is an isosceles right triangle.

If one leg measures 24 units, the other leg is 24 units.

The length of the hypotenuse is $24 \sqrt{2}$ units.

In a right triangle, the altitude to the hypotenuse divides the triangle into two similar triangles to the original triangle.
In any right triangle, the altitude to the hypotenuse is the geometric mean between the segments into which it divides the hypotenuse. To illustrate, we have the equation $\frac{B D}{A D} = \frac{A D}{C D}$ .
Each leg is a geometric mean between the hypotenuse and the hypotenuse segment adjacent to the leg. To illustrate, we have the following equations: $\frac{B C}{A B} = \frac{A B}{B D}$ and $\frac{B C}{A C} = \frac{A C}{C D}$

In the 30-60-90 Triangle, if the longer leg $a$ has a measure $x$ and the hypotenuse is denoted by $c$ , then $c = 2 x$ and $b = x \sqrt{3}$ units.
In the 45-45-90 Triangle, if one leg measures $x$ , then the other leg measures $x$ and the hypotenuse measures $x \sqrt{2}$ units.
The side opposite the 30-degree angle is the shorter leg.
The side opposite the 60-degree angle is the longer leg.
The angle opposite the shorter leg measures $30 °$ .
The angle opposite the longer leg measures $60 °$ .

The sides of a right triangle are not closed using the symbols $a, b,$ or $c$ . Any letter symbol can be used as long as the concepts of the theorems are appropriately utilized.
It is typical that uppercase letters are used to denote angles while lowercase letters are used to denote side lengths of a right triangle.
30-60-90 Triangle Theorem is used to derive the formula of the Area of an Equilateral Triangle, $A = \frac{s^{2} \sqrt{3}}{4}$ .
The Isosceles Triangle Theorem and Median Theorem of a right triangle can be used in proving the 30-60-90 Triangle Theorem.
Pythagorean Theorem can be used in proving the 45-45-90 Triangle Theorem.