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# 30-60-90 Right Triangles

## Hypotenuse equals twice the smallest leg, while the larger leg is sqrt(3) times the smallest.

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30-60-90 Triangles

Have you ever tried to measure a flagpole? How about a shadow? Take a look at this dilemma.

The diagram below shows the shadow a flagpole casts at a certain time of day. If the height of the flagpole is \begin{align*}7 \sqrt{3} \ feet\end{align*}, what is the length of the hypotenuse of the triangle formed by the flagpole and its shadow?

This Concept will show you how to work with special triangle relationships and solve this dilemma.

### Guidance

A special type of triangle has angles measuring \begin{align*}30^\circ,60^\circ\end{align*}, and \begin{align*}90^\circ\end{align*}. These triangles are exactly one half of equilateral triangles.

Do you remember what an equilateral triangle is?

An equilateral triangle is a triangle with equal angle measures.

The equal angle measures of an equilateral triangle are \begin{align*}60^\circ-60^\circ-60^\circ\end{align*}. If we divide an equilateral triangle in half, then we have a \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle.

You can tell in the diagram that since the original triangle was equilateral, the short leg will be one-half the length of the hypotenuse.

Take a look at this situation.

Find the length of the missing leg in the triangle below. Use the Pythagorean theorem to find your answer.

Just like you did for \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} triangles, use the Pythagorean Theorem to find the missing side. In this diagram, you are given two measurements. The hypotenuse \begin{align*}(c)\end{align*} is 2 feet and the shorter leg \begin{align*}(a)\end{align*} is 1 foot. Find the length of the missing leg \begin{align*}(b)\end{align*}.

You can leave the answer as the radical as shown, or use your calculator to find the approximate value of 1.732 feet.

So, just as there is a constant proportion relating the sides of the \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} triangle, there is also one relating the sides of the \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle. The hypotenuse will always be twice the length of the shorter leg, and the longer leg is always the product of the shorter leg and \begin{align*}\sqrt{3}\end{align*}.

Write this information down under 30/60/90 degree right triangles.

You can also use this information when problem solving. Take a look.

What is the length of one leg in the triangle below?

The first step in this problem is to identify the type of right triangle depicted. The angles show that this is a \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle. So, the longer leg is the product of one leg and \begin{align*}\sqrt{3}\end{align*}. The hypotenuse is twice the length of the shorter leg. The shorter leg is 4 meters.

The hypotenuse will be \begin{align*}4 \times 2\end{align*}, or 8 meters long.

Problem solve using what you have learned about special right triangles.

#### Example A

What is the length of the hypotenuse if the shorter leg is 6 units?

Solution: 12 units

#### Example B

What is the length of the longer leg if the shorter leg is 5 units?

Solution: \begin{align*}5 \sqrt{3}\end{align*}

#### Example C

If the length of the hypotenuse is 14, what is the length of the longer side?

Solution: \begin{align*}7 \sqrt{3}\end{align*}

Now let's go back to the dilemma from the beginning of the Concept.

The wording in this problem is complicated, but you only need to notice a few things. You can tell in the picture that this triangle has angles of \begin{align*}30^\circ,60^\circ\end{align*}, and \begin{align*}90^\circ\end{align*}. The actual flagpole is the longer leg in the triangle, so use the ratios in the diagrams above to find the length of the hypotenuse.

The longer leg is the product of the shorter leg and \begin{align*}\sqrt{3}\end{align*}. So, the length of the shorter leg will be 7 feet.

The hypotenuse in a \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle will always be twice the length of the shorter leg, so it will equal \begin{align*}7 \times 2\end{align*}, or 14 feet.

### Vocabulary

Equilateral Triangle
a triangle with all three angles \begin{align*}60^\circ\end{align*}.
30/60/90 Triangle
a special right triangle that is created when an equilateral triangle is divided in half.

### Guided Practice

Here is one for you to try on your own.

What is the length of the missing leg in the a 30/60/90 degree right triangle with a short leg of 5 and a hypotenuse of 10?

Solution

Since the length of the longer leg is the product of the shorter leg and \begin{align*}\sqrt{3}\end{align*}, you can easily calculate this length. The short leg is 5 inches, so the hypotenuse will be \begin{align*}5 \sqrt{3}\end{align*}, or about 8.66 inches.

### Practice

Directions: Find the missing length of the longer leg in each \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangle.

1. short leg = 3
2. short leg = 4
3. short leg = 2
4. short leg = 8
5. short leg = 10

Directions: Now use a calculator to figure out the approximate value of each longer leg. You may round your answer when necessary.

1. \begin{align*}3 \sqrt{3}\end{align*}
2. \begin{align*}4 \sqrt{3}\end{align*}
3. \begin{align*}2 \sqrt{3}\end{align*}
4. \begin{align*}8 \sqrt{3}\end{align*}
5. \begin{align*}10 \sqrt{3}\end{align*}

Directions: Use what you have learned to solve each problem.

1. Janie had construction paper cut into and equilateral triangle. She wants to cut it into two smaller congruent triangles. What will be the angle measurement of the triangles that result?
2. Madeleine has poster board in the shape of a square. She wants to cut two congruent triangles out of the poster board without leaving any leftovers. What will be the angle measurements of the triangles that result?
3. A square window has a diagonal of \begin{align*}2 \sqrt{2} \ feet\end{align*}. What is the length of the shorter of its sides?
4. A square block of cheese is cut into two congruent wedges. If the shortest side of the original block was 9 inches, how long is the diagonal cut?
5. Jerry wants to find the area of an equilateral triangle but only knows that the length of the shorter side is 4 centimeters. What is the height of Jerry’s triangle?

### Vocabulary Language: English

30-60-90 Theorem

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
30-60-90 Triangle

30-60-90 Triangle

A 30-60-90 triangle is a special right triangle with angles of $30^\circ$, $60^\circ$, and $90^\circ$.
Equilateral Triangle

Equilateral Triangle

An equilateral triangle is a triangle in which all three sides are the same length.
Hypotenuse

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

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

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.
The $\sqrt{}$, or square root, sign.