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

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Identify and use the ratios involved with 306090\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 60\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 90\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 60\begin{align*}60^\circ \end{align*} angle in the picture in half. Therefore, the angles on either side of the altitude are 30\begin{align*}30^\circ\end{align*} (because 60÷2=30\begin{align*}60^\circ \div 2 = 30^\circ\end{align*}).

Each resulting right triangle created is a 306090\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 90\begin{align*}90^\circ\end{align*} angle at the base and splits the 60\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 306090\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 306090\begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle is always double the length of the side opposite the 30\begin{align*}30^\circ\end{align*} angle. You can use this information to solve problems about equilateral triangles.

In a 306090\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 306090\begin{align*}30^\circ - 60^\circ - 90^\circ\end{align*} triangle, the side _______________ the 30\begin{align*}30^\circ\end{align*} angle is the shortest side.

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

This important type of right triangle has angles measuring 30,60,\begin{align*}30^\circ, 60^\circ,\end{align*} and 90\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 for454590\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 (a2+b2=c2)\begin{align*}(a^2+ b^2 = c^2)\end{align*} to find the length of the missing leg (b)\begin{align*}( b )\end{align*}:

a2+b2=  c2   12+b2=  22 1+b2=  41   1   b2=  3 b=  3\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 b1.732 cm\begin{align*}b\approx1.732 \ cm\end{align*}.

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

Recall that since the 306090\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, c\begin{align*}c\end{align*}, is 6 feet.

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

a2+b2 = c232+b2 = 62 9+b2 = 369   9b2 = 27 b = 27   = 9×3=33 ft5.196 ft\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 306090\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 3\begin{align*}\sqrt{3}\end{align*}.

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

x:x3:2x\begin{align*}x:x\sqrt{3}:2x\end{align*}

If we use ratio form, where the ratio is x:x3:2x\begin{align*} x:x\sqrt{3}:2x\end{align*}

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

For instance, if x=4\begin{align*} x = 4\end{align*}, the ratio is 4:43:2(4)\begin{align*} 4 : 4\sqrt{3} : 2(4)\end{align*} or 4:43:8\begin{align*} 4 : 4\sqrt{3}: 8\end{align*} and if x=7\begin{align*} x = 7\end{align*}, the ratio is 7:73:2(7)\begin{align*}7 : 7\sqrt{3} : 2(7)\end{align*} or 7:73:14\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 306090\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 3\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 60\begin{align*}60^\circ\end{align*} angle.

Since the length of the longer leg is the product of the shorter leg and 3\begin{align*}\sqrt{3}\end{align*}, you can easily calculate this length:

The short leg is 8 inches, so the longer leg will be 83\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 AC¯¯¯¯¯¯¯¯\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 30,60\begin{align*}30^\circ, 60^\circ\end{align*}, and 90.\begin{align*}90^\circ.\end{align*}

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

AC=BC3=43\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.

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.

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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