1.2: Special Right Triangles

Learning Objectives

• Recognize Special Right Triangles.
• Use the special right triangle rations to solve special right triangles.

30-60-90 Right Triangles

Hypotenuse equals twice the smallest leg, while the larger leg is $\sqrt{3}$ times the smallest.

One of the two special right triangles is called a 30-60-90 triangle, after its three angles.

30-60-90 Theorem: If a triangle has angle measures $30^{\circ}$, $60^{\circ}$ and $90^{\circ}$, then the sides are in the ratio $x: x\sqrt{3}:2x$.

The shorter leg is always $x$, the longer leg is always $x\sqrt{3}$, and the hypotenuse is always $2x$. If you ever forget these theorems, you can still use the Pythagorean Theorem.

What if you were given a 30-60-90 right triangle and the length of one of its side? How could you figure out the lengths of its other sides?

Example $\PageIndex{1}$

Find the value of $x$ and $y$.

Figure $\PageIndex{1}$

Solution

We are given the longer leg.

$\begin{aligned} x\sqrt{3} &=12 \\ x&=12\sqrt{3}\cdot \dfrac{\sqrt{3}}{\sqrt{3}}=12\dfrac{\sqrt{3}}{3}=4\sqrt{3} \\ &\text{The hypotenuse is} \\ y&=2(4\sqrt{3})=8\sqrt{3} \end{aligned}$

Example $\PageIndex{2}$

Find the value of $x$ and $y$.

Figure $\PageIndex{2}$

Solution

We are given the hypotenuse.

$\begin{aligned} 2x&=16 \\ x&=8 \\ \text{The longer leg is} \\ y&=8\cdot \sqrt{3}&=8\sqrt{3} \end{aligned}$

Example $\PageIndex{3}$

Find the length of the missing sides.

Figure $\PageIndex{3}$

Solution

We are given the shorter leg. If $x=5$, then the longer leg, $b=5\sqrt{3}$, and the hypotenuse, $c=2(5)=10$.

Example $\PageIndex{4}$

Find the length of the missing sides.

Figure $\PageIndex{4}$

Solution

We are given the hypotenuse. $2x=20$, so the shorter leg, $f=\dfrac{20}{2}=10$, and the longer leg, $g=10\sqrt{3}$.

Example $\PageIndex{5}$

A rectangle has sides 4 and $4\sqrt{3}$. What is the length of the diagonal?

Solution

If you are not given a picture, draw one.

Figure $\PageIndex{5}$

The two lengths are $x$, $x\sqrt{3}$, so the diagonal would be $2x$, or $2(4)=8$.

If you did not recognize this is a 30-60-90 triangle, you can use the Pythagorean Theorem too.

$\begin{aligned} 4^2+(4\sqrt{3})^2&=d^2 \\ 16+48&=d^2 \\ d=\sqrt{64}&=8 \end{aligned}$

Review

1. In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
2. In a 30-60-90 triangle, if the shorter leg is $x$, then the longer leg is __________ and the hypotenuse is ___________.
3. A rectangle has sides of length 6 and $6\sqrt{3}$. What is the length of the diagonal?
4. Two (opposite) sides of a rectangle are 10 and the diagonal is 20. What is the length of the other two sides

45-45-90 Right Triangles

A right triangle with congruent legs and acute angles is an Isosceles Right Triangle. This triangle is also called a 45-45-90 triangle (named after the angle measures).

Figure $\PageIndex{6}$

$\Delta ABC$ is a right triangle with $m\angle A=90^{\circ}$, $\overline{AB} \cong \overline{AC}$ and $m\angle B=m\angle C=45^{\circ}$.

45-45-90 Theorem: If a right triangle is isosceles, then its sides are in the ratio $x:x:x\sqrt{2}$. For any isosceles right triangle, the legs are $x$ and the hypotenuse is always $x\sqrt{2}$.

What if you were given an isosceles right triangle and the length of one of its sides? How could you figure out the lengths of its other sides?

Example $\PageIndex{6}$

Find the length of $x$.

Solution

Use the $x:x:x\sqrt{2}$ ratio.

Here, we are given the hypotenuse. Solve for $x$ in the ratio.

$\begin{aligned} x\sqrt{2} =16\\ x=16\sqrt{2}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}=\dfrac{16\sqrt{2}}{2}=8\sqrt{2} \end{aligned}$

Example $\PageIndex{7}$

Find the length of $x$, where $x$ is the hypotenuse of a 45-45-90 triangle with leg lengths of $5\sqrt{3}$.

Solution

Use the $x:x:x\sqrt{2}$ ratio.

$x=5\sqrt{3}\cdot\sqrt{2}=5\sqrt{6}$

Example $\PageIndex{8}$

Find the length of the missing side.

Figure $\PageIndex{8}$

Solution

Use the $x:x:x\sqrt{2}$ ratio. $TV=6$ because it is equal to $ST$. So, $SV=6 \cdot \sqrt{2}=6\sqrt{2}$.

Example $\PageIndex{9}$

Find the length of the missing side.

Figure $\PageIndex{9}$

Solution

Use the $x:x:x\sqrt{2}$ ratio. $AB=9\sqrt{2}$ because it is equal to $AC$. So, $BC=9\sqrt{2}\cdot\sqrt{2}=9\cdot 2=18$.

Example $\PageIndex{10}$

A square has a diagonal with length 10, what are the lengths of the sides?

Solution

Draw a picture.

We know half of a square is a 45-45-90 triangle, so $10=s\sqrt{2}$.

$\begin{aligned} s\sqrt{2}&=10 \\ s&=10\sqrt{2}\cdot \dfrac{\sqrt{2}}{\sqrt{2}}=\dfrac{10\sqrt{2}}{2}=5\sqrt{2} \end{aligned}$

Review

1. In an isosceles right triangle, if a leg is 4, then the hypotenuse is __________.
2. In an isosceles right triangle, if a leg is x, then the hypotenuse is __________.
3. A square has sides of length 15. What is the length of the diagonal?
4. A square’s diagonal is 22. What is the length of each side?

Resources

Vocabulary

Term	Definition
30-60-90 Theorem	If a triangle has angle measures of 30, 60, and 90 degrees, then the sides are in the ratio $x : x \sqrt{3} : 2x$
45-45-90 Theorem	For any isosceles right triangle, if the legs are x units long, the hypotenuse is always $x\sqrt{2}$.
Hypotenuse	The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.
Legs of a Right Triangle	The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.
Pythagorean Theorem	The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by $a^2+b^2=c^2$, where a and b are legs of the triangle and c is the hypotenuse of the triangle.
Radical	The $\sqrt{}$, or square root, sign.

Additional Resources

Interactive Element

Video: Solving Special Right Triangles

Activities: 30-60-90 Right Triangles Discussion Questions

Study Aids: Special Right Triangles Study Guide

Practice: 30-60-90 Right Triangles 45-45-90 Right Triangles

Real World: Fighting the War on Drugs Using Geometry and Special Triangles