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# 8.5: 30-60-90 Right Triangles

Difficulty Level: At Grade Created by: CK-12
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Practice 30-60-90 Right Triangles

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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? After completing this Concept, you'll be able to use the 30-60-90 Theorem to solve problems like this one.

### Watch This

Watch the first half of this video.

Now watch the first half of this video.

### Guidance

One of the special right triangles is called a 30-60-90 triangle, after its three angles. To construct a 30-60-90 triangle, start with an equilateral triangle.

##### Investigation: Properties of a 30-60-90 Triangle

Tools Needed: Pencil, paper, ruler, compass

1. Construct an equilateral triangle with 2 in sides.

2. Draw or construct the altitude from the top vertex to the base for two congruent triangles.

3. Find the measure of the two angles at the top vertex and the length of the shorter leg.

The top angles are each 30\begin{align*}30^\circ\end{align*} and the shorter leg is 1 in because the altitude of an equilateral triangle is also the angle and perpendicular bisector.

4. Find the length of the longer leg, using the Pythagorean Theorem. Simplify the radical.

5. Now, let’s say the shorter leg is length x\begin{align*}x\end{align*} and the hypotenuse is 2x\begin{align*}2x\end{align*}. Use the Pythagorean Theorem to find the longer leg. What is it? How is this similar to your answer in #4?

x2+b2x2+b2b2b=(2x)2=4x2=3x2=x3\begin{align*}x^2 + b^2 & = (2x)^2\\ x^2 + b^2 & = 4x^2\\ b^2 & = 3x^2\\ b & = x \sqrt{3}\end{align*}

30-60-90 Corollary: If a triangle is a 30-60-90 triangle, then its sides are in the extended ratio x:x3:2x\begin{align*}x : x \sqrt{3} : 2x\end{align*}.

Step 5 in the above investigation proves the 30-60-90 Corollary. The shortest leg is always x\begin{align*}x\end{align*}, the longest leg is always x3\begin{align*}x \sqrt{3}\end{align*}, and the hypotenuse is always 2x\begin{align*}2x\end{align*}. If you ever forget this corollary, then you can still use the Pythagorean Theorem.

#### Example A

Find the length of the missing sides.

We are given the shortest leg. If x=5\begin{align*}x = 5\end{align*}, then the longer leg, b=53\begin{align*}b = 5 \sqrt{3}\end{align*}, and the hypotenuse, c=2(5)=10\begin{align*}c = 2(5) = 10\end{align*}.

#### Example B

Find the value of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}.

We are given the longer leg.

x3xxx=12=12333=1233=43\begin{align*}x \sqrt{3} &= 12\\ x &= \frac{12}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \\ x&= \frac{12 \sqrt{3}}{3} \\x&= 4 \sqrt{3}\end{align*}

Then, the hypotenuse would be y=2(43)=83\begin{align*}y = 2 \left ( 4 \sqrt{3} \right ) = 8 \sqrt{3}\end{align*}.

#### Example C

Find the measure of x\begin{align*}x\end{align*}.

Think of this trapezoid as a rectangle, between a 45-45-90 triangle and a 30-60-90 triangle.

From this picture, x=a+b+c\begin{align*}x = a + b + c\end{align*}. First, find a\begin{align*}a\end{align*}, which is a leg of an isosceles right triangle.

a=24222=2422=122\begin{align*}a = \frac{24}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{24 \sqrt{2}}{2} = 12 \sqrt{2}\end{align*}

a=d\begin{align*}a = d\end{align*}, so we can use this to find c\begin{align*}c\end{align*}, which is the shorter leg of a 30-60-90 triangle.

c=122333=1263=46\begin{align*}c = \frac{12 \sqrt{2}}{ \sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{12 \sqrt{6}}{3} = 4 \sqrt{6}\end{align*}

b=20\begin{align*}b = 20\end{align*}, so x=122+20+46\begin{align*}x = 12 \sqrt{2} + 20 + 4 \sqrt{6}\end{align*}. Nothing simplifies, so this is how we leave our answer.

Watch this video for help with the Examples above.

### Vocabulary

A right triangle is a triangle with a 90\begin{align*}90^\circ\end{align*} angle. A 30-60-90 triangle is a right triangle with angle measures of 30,60\begin{align*}30^\circ, 60^\circ\end{align*}, and 90\begin{align*}90^\circ\end{align*}.

### Guided Practice

1. Find the length of the missing sides.

2. Find the value of x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*}.

3. x\begin{align*}x\end{align*} is the hypotenuse of a 30-60-90 triangle and y\begin{align*}y\end{align*} is the longer leg of the same triangle. The shorter leg has a length of 6\begin{align*}6\end{align*}.

1. We are given the hypotenuse. 2x=20\begin{align*}2x = 20\end{align*}, so the shorter leg, f=10\begin{align*}f = 10\end{align*}, and the longer leg, g=103\begin{align*}g = 10 \sqrt{3}\end{align*}.

2. We are given the hypotenuse.

2xx=156=1562\begin{align*}2x &= 15 \sqrt{6}\\ x &= \frac{15 \sqrt{6}}{2}\\ \end{align*}

The, the longer leg would be y=(1562)3=15182=4522\begin{align*}y = \left ( \frac{15 \sqrt{6}}{2} \right ) \cdot \sqrt{3} = \frac{15 \sqrt{18}}{2} = \frac{45 \sqrt{2}}{2}\end{align*}

3. We are given the shorter leg.

x=2(6)x=12The longer leg isy=63=63\begin{align*}& x=2(6)\\ & x = 12\\ & \text{The longer leg is}\\ &y= 6 \cdot \sqrt 3 = 6 \sqrt{3}\end{align*}

### Practice

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\begin{align*}x\end{align*}, then the longer leg is __________ and the hypotenuse is ___________.
3. A rectangle has sides of length 6 and 63\begin{align*}6 \sqrt{3}\end{align*}. 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?

For questions 5-12, find the lengths of the missing sides. Simplify all radicals.

1. What is the height of an equilateral triangle with sides of length 3 in?
2. What is the area of an equilateral triangle with sides of length 5 ft?
3. A regular hexagon has sides of length 3 in. What is the area of the hexagon? (Hint: the hexagon is made up a 6 equilateral triangles.)
4. The area of an equilateral triangle is 363\begin{align*}36 \sqrt{3}\end{align*}. What is the length of a side?
5. If a road has a grade of 30\begin{align*}30^\circ\end{align*}, this means that its angle of elevation is 30\begin{align*}30^\circ\end{align*}. If you travel 1.5 miles on this road, how much elevation have you gained in feet (5280 ft = 1 mile)?

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### Vocabulary Language: English

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

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

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.

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

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