# 3.12: Special Right Triangles, 30-60-90

**At Grade**Created by: CK-12

## Learning Objectives

- Identify and use the ratios involved with \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangles.
- Identify and use ratios involved with equilateral triangles.

## Equilateral Triangles

Remember that an **equilateral** triangle has sides that all have the same length. **Equilateral** triangles are also **equiangular** — all angles have the same measure. In an equilateral triangle, all angles measure exactly \begin{align*}60^\circ\end{align*}.

Equilateral triangles are also ___________________.

Notice what happens when you divide an equilateral triangle in half:

This equilateral triangle is divided into 2 equal parts using an **altitude,** which is a line that is *perpendicular* to the base of the triangle. Since the altitude is perpendicular to the base, it makes a \begin{align*}90^\circ\end{align*} angle with the base.

- An altitude is a line that is ______________________________ to the base of a triangle.

The altitude also splits the top \begin{align*}60^\circ \end{align*} angle in the picture in half. Therefore, the angles on either side of the altitude are \begin{align*}30^\circ\end{align*} (because \begin{align*}60^\circ \div 2 = 30^\circ\end{align*}).

Each resulting right triangle created is a \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle:

- The hypotenuse of the resulting triangle is the side of the original, and the shorter leg is half of an original side.
- The
**altitude**makes a \begin{align*}90^\circ\end{align*} angle at the base and splits the \begin{align*}60^\circ\end{align*} angle into two __________ angles.

This is why the **hypotenuse** is always *twice the length* of the *shorter* **leg** in a \begin{align*}30^\circ - 60^\circ - 90^\circ \end{align*} triangle, like in the picture below (where the original equilateral triangle had a side of length 10):

Again, the **hypotenuse** of a \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle is *always* double the length of the side *opposite* the \begin{align*}30^\circ\end{align*} angle. You can use this information to solve problems about equilateral triangles.

In a \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle, the_____________ is *double* the shortest side.

The ________________ is the longest side of any right triangle.

In a \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle, the side _______________ the \begin{align*}30^\circ\end{align*} angle is the shortest side.

## \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} Triangles

This important type of right triangle has angles measuring \begin{align*}30^\circ, 60^\circ,\end{align*} and \begin{align*}90^\circ\end{align*}.

Just as you found a constant ratio between the sides of an isosceles right triangle, you can find constant ratios here as well. Use the Pythagorean Theorem to discover these important relationships.

**Example 1**

*Find the length of the missing leg in the following triangle. 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**(which is side ___________ ) is*2 cm*and - The shorter
**leg**(which is side ___________ ) is*1 cm*

Substitute these values into the Pythagorean Theorem \begin{align*}(a^2+ b^2 = c^2)\end{align*} to find the length of the missing leg \begin{align*}( b )\end{align*}:

\begin{align*}& \ \ \ a^2+ b^2 = \ \ c^2\\ & \ \ \ 1^2+ b^2 = \ \ 2^2\\ & \ \quad 1+ b^2 = \ \ 4\\ & - 1 \qquad \ \ \ -1\\ & \ \ \ \qquad b^2 = \ \ 3\\ & \ \qquad \quad b = \ \ \sqrt{3}\end{align*}

You can leave the answer in radical form as shown, or use your calculator to find the approximate value of \begin{align*}b\approx1.732 \ cm\end{align*}.

We can try this again using a hypotenuse of 6 feet.

Recall that since the \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle comes from an **equilateral** triangle, you know that the length of the *shorter* **leg** is *half* the length of the **hypotenuse.**

So the **hypotenuse**, \begin{align*}c\end{align*}, is 6 feet.

Therefore, the shorter leg, \begin{align*}a\end{align*} in the diagram on the previous page, is 3 feet (half of 6.)

\begin{align*}& \quad a^2+b^2 \ = \ c^2\\ & \quad 3^2+b^2 \ = \ 6^2\\ & \quad \ 9+b^2 \ = \ 36\\ & - 9 \qquad \ \ \ -9\\ & \qquad \quad b^2 \ = \ 27\\ & \qquad \ \quad b \ = \ \sqrt{27}\\ & \qquad \ \ \quad \ = \ \sqrt{9} \times \sqrt{3} = 3\sqrt{3} \ ft \approx 5.196 \ ft\end{align*}

The special relationship is as follows:

In all \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangles,

- the hypotenuse will always be
**twice**the length of the shorter leg, - and the longer leg is always the product of the length of the shorter leg and \begin{align*}\sqrt{3}\end{align*}.

In ratio form, the sides, in order from *shortest* to *longest* are in the extended ratio

\begin{align*}x:x\sqrt{3}:2x\end{align*}

*If we use ratio form, where the ratio is* \begin{align*} x:x\sqrt{3}:2x\end{align*}

*We can substitute any number in for \begin{align*}x\end{align*} and the sides of the triangle will relate to each other in the same proportion.*

*For instance, if* \begin{align*} x = 4\end{align*}, *the ratio is* \begin{align*} 4 : 4\sqrt{3} : 2(4)\end{align*} *or* \begin{align*} 4 : 4\sqrt{3}: 8\end{align*} *and if* \begin{align*} x = 7\end{align*}, *the ratio is* \begin{align*}7 : 7\sqrt{3} : 2(7)\end{align*} *or* \begin{align*}7 : 7\sqrt{3}: 14\end{align*}

**Example 2**

*What is the length of the missing leg in the triangle below?*

You could use the Pythagorean Theorem for this problem (like in Example 1), but using the special proportional relationship in a \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle that you just learned is a much easier way!

The special relationship is:

- the
**hypotenuse**is _____________________ the length of the*shorter***leg**, and - the
*longer***leg**is the _______________________ of the length of the*shorter***leg**and \begin{align*}\sqrt{3}\end{align*}

First, you know that the **hypotenuse** is 16 because it is *across* from the **right** angle.

Therefore, the other 2 sides are the **legs** in this triangle.

You also know that the *longer* **leg** is across from the \begin{align*}60^\circ\end{align*} angle.

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 *8 inches,* so the longer leg will be \begin{align*}8\sqrt{3}\end{align*} *inches* or about 13.86 inches.

**Example 3**

*What is AC below?*

To find the length of segment \begin{align*}\overline {AC}\end{align*}, identify its relationship to the rest of the triangle. Since it is an **altitude**, it forms two congruent triangles with angles measuring \begin{align*}30^\circ, 60^\circ\end{align*}, and \begin{align*}90^\circ.\end{align*}

So, \begin{align*}AC\end{align*} will be the product of \begin{align*}BC\end{align*} (the shorter leg) and \begin{align*}\sqrt{3}\end{align*}:

\begin{align*} AC & = BC \sqrt{3}\\ & = 4 \sqrt{3}\end{align*}

\begin{align*}AC = 4\sqrt{3}\end{align*} yards, or approximately 6.93 yards.

**Reading Check:**

1. *Draw a* \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} *triangle and label its angle measures.*

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

\begin{align*}{\;}\end{align*}

2. *From shortest to longest sides, what ratio does every* \begin{align*} 30^\circ - 60^\circ - 90^\circ\end{align*} *triangle follow?*

________ : ________ : ________

**Example 4**

*The diagram below shows the shadow a flagpole casts at a certain time of day.*

*If the length of the shadow cast by the flagpole is 13m, what is the height of the flagpole and what is the length of the hypotenuse of the right triangle shown?*

The picture shows that this triangle has angles of \begin{align*} 30^\circ, 60^\circ,\end{align*} and \begin{align*}90^\circ\end{align*} (This assumes that the flagpole is perpendicular to the ground, but that is a safe assumption). Although the \begin{align*}30^\circ\end{align*} angle is not written into the picture, you can tell that the top angle is \begin{align*}30^\circ\end{align*} because \begin{align*}180 - (90 + 60) = 30.\end{align*}

The height of the flagpole is the *longer* leg in the triangle, so use the special right triangle ratios (along with the given height of the base of the triangle) to find the length of the missing sides, the flagpole height and the hypotenuse.

The *longer* leg is the *product* of the *shorter* leg and \begin{align*}\sqrt{3}\end{align*}.

The length of the shorter leg is given as 13 meters, so the height of the flagpole is \begin{align*}13\sqrt{3} \ m.\end{align*}

To find the length of the *hypotenuse,* use the hypotenuse of a \begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle.

It will always be **twice** the length of the shorter leg, so it will equal \begin{align*}13 \cdot 2\end{align*}, or 26 meters.

**Reading Check:**

*What is the length of the* *altitude**in the triangle below?*

## Graphic Organizer for Lessons 9 and 10

**Special Triangles**

*Label all angles and side lengths for the triangles below. If the shortest side of each triangle is length* \begin{align*}x\end{align*}, *what are the other sides?*

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