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# Relationships of Sides in 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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Practice Relationships of Sides in 30-60-90 Right Triangles
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Relationships of Sides in 30-60-90 Right Triangles

You are working on a project in your Industrial Arts class. You have been instructed to design a building. You have a plastic triangular piece that helps you with straight edges and designs. This triangle has interior angles of 30\begin{align*}30^\circ\end{align*}, 60\begin{align*}60^\circ\end{align*}, and 90\begin{align*}90^\circ\end{align*}. Laying the triangle flat on the paper, you decide to use it to help you with the right angle at the building's base.

You trace out the bottom and left edge of the triangle on the paper to serve as the side and bottom of the structure. Looking at the length of the base of the building, you see that it is 7 inches long. Can you determine what the height of the building is from this information?

### 30-60-90 Right Triangles

306090\begin{align*}30-60-90\end{align*} refers to each of the angles in this special right triangle.

To understand the ratios of the sides, start with an equilateral triangle with an altitude drawn from one vertex.

Recall from geometry that an altitude, h\begin{align*}h\end{align*}, cuts the opposite side directly in half. So we know that one side, the hypotenuse, is 2s\begin{align*}2s\end{align*} and the shortest leg is s\begin{align*}s\end{align*}. Also, recall that the altitude is a perpendicular and angle bisector, which is why the angle at the top is split in half. To find the length of the longer leg, use the Pythagorean Theorem:

s2+h2s2+h2h2h=(2s)2=4s2=3s2=s3\begin{align*}s^2 + h^2 & = (2s)^2\\ s^2 + h^2 & = 4s^2\\ h^2 & = 3s^2\\ h & = s\sqrt{3}\end{align*}

From this we can conclude that the length of the longer leg is the length of the short leg multiplied by 3\begin{align*}\sqrt{3}\end{align*} or s3\begin{align*}s\sqrt{3}\end{align*}. Just like the isosceles right triangle, we now only need one side in order to determine the other two in a 306090\begin{align*}30-60-90\end{align*} triangle. The ratio of the three sides is written x:x3:2x\begin{align*}x:x\sqrt{3}:2x\end{align*}, where x\begin{align*}x\end{align*} is the shortest leg, x3\begin{align*}x\sqrt{3}\end{align*} is the longer leg and 2x\begin{align*}2x\end{align*} is the hypotenuse.

Notice, that the shortest side is always opposite the smallest angle and the longest side is always opposite 90\begin{align*}90^\circ\end{align*}.

#### Finding the Length of Missing Sides

1. Find the lengths of the two missing sides in the 306090\begin{align*}30-60-90\end{align*} triangle.

Determine which side in the 306090\begin{align*}30-60-90\end{align*} ratio is given and solve for the other two.

43\begin{align*}4\sqrt{3}\end{align*} is the longer leg because it is opposite the 60\begin{align*}60^\circ\end{align*}. So, in the x:x3:2x\begin{align*}x:x\sqrt{3}:2x\end{align*} ratio, 43=x3\begin{align*}4\sqrt{3} = x\sqrt{3}\end{align*}, therefore x=4\begin{align*}x = 4\end{align*} and 2x=8\begin{align*}2x = 8\end{align*}. The short leg is 4 and the hypotenuse is 8.

2. Find the lengths of the two missing sides in the 306090\begin{align*}30-60-90\end{align*} triangle.

Determine which side in the 306090\begin{align*}30-60-90\end{align*} ratio is given and solve for the other two.

17 is the hypotenuse because it is opposite the right angle. In the x:x3:2x\begin{align*}x:x\sqrt{3}:2x\end{align*} ratio, 17=2x\begin{align*}17 = 2x\end{align*} and so the short leg is 172\begin{align*}\frac{17}{2}\end{align*} and the long leg is 1732\begin{align*}\frac{17\sqrt{3}}{2}\end{align*}.

3. Find the lengths of the two missing sides in the 306090\begin{align*}30-60-90\end{align*} triangle.

Determine which side in the 306090\begin{align*}30-60-90\end{align*} ratio is given and solve for the other two.

15 is the long leg because it is opposite the 60\begin{align*}60^\circ\end{align*}. Even though 15 does not have a radical after it, we can still set it equal to x3\begin{align*}x\sqrt{3}\end{align*}.

x3x=15=15333=1533=53So, the short leg is 53.\begin{align*}x\sqrt{3} & = 15\\ x & = \frac{15}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{15\sqrt{3}}{3} = 5\sqrt{3} \quad \text{So, the short leg is} \ 5\sqrt{3}.\end{align*}

Multiplying 53\begin{align*}5\sqrt{3}\end{align*} by 2, we get the hypotenuse length, which is 103\begin{align*}10\sqrt{3}\end{align*}.

### Examples

#### Example 1

Earlier, you were asked to design a building.

Since you know the ratios of sides of a 30-60-90 triangle, you know that since the bottom side has a length of 7\begin{align*}7\end{align*} cm, the left side must have a length of 7312.12\begin{align*}7\sqrt{3} \approx 12.12\end{align*} cm.

#### Example 2

Find the lengths of the two missing sides in the 306090\begin{align*}30-60-90\end{align*} triangle.

Since the given side has a length of 8 and is the side which is opposite the right angle, we know that this is the "2x" side of the triangle. Therefore, the short side of the triangle is 82=4\begin{align*}\frac{8}{2} = 4\end{align*} and the third side of the triangle is 43\begin{align*}4 \sqrt{3}\end{align*}.

#### Example 3

Find the lengths of the two missing sides in the 30-60-90 triangle below.

We can see that the shortest side of the triangle has a length of 3 while the longest side has a length of 6 and the other side has a length of 33\begin{align*}3 \sqrt{3}\end{align*}. This means that the 30\begin{align*}30^\circ\end{align*} angle is opposite the side with length 3\begin{align*}3\end{align*}, the 60\begin{align*}60^\circ\end{align*} angle is opposite the side with length 33\begin{align*}3 \sqrt{3}\end{align*}, and the 90\begin{align*}90^\circ\end{align*} angle is opposite the side with length 6\begin{align*}6\end{align*}.

#### Example 4

Find the angles in the 306090\begin{align*}30-60-90\end{align*} triangle.

The length of the given side is 33\begin{align*}3\sqrt{3}\end{align*}, and is opposite the 60\begin{align*}60^\circ\end{align*} angle. This means that the side opposite the 30\begin{align*}30^\circ\end{align*} angle is 3\begin{align*}3\end{align*}, and the length of the side opposite the 90\begin{align*}90^\circ\end{align*} angle is 6\begin{align*}6\end{align*}

### Review

1. In a 30-60-90 triangle, if the shorter leg is 8, then the longer leg is __________ and the hypotenuse is ___________.
2. In a 30-60-90 triangle, if the shorter leg is 12, then the longer leg is __________ and the hypotenuse is ___________.
3. In a 30-60-90 triangle, if the longer leg is 10, then the shorter leg is __________ and the hypotenuse is ___________.
4. In a 30-60-90 triangle, if the shorter leg is 16, then the longer leg is __________ and the hypotenuse is ___________.
5. In a 30-60-90 triangle, if the longer leg is 3, then the shorter leg is __________ and the hypotenuse is ___________.
6. 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 ___________.
7. In a 30-60-90 triangle, if the longer leg is x\begin{align*}x\end{align*}, then the shorter leg is __________ and the hypotenuse is ___________.
8. A rectangle has sides of length 7 and 73\begin{align*}7 \sqrt{3}\end{align*}. What is the length of the diagonal?
9. Two (opposite) sides of a rectangle are 15 and the diagonal is 30. What is the length of the other two sides?
10. What is the height of an equilateral triangle with sides of length 6 in?
11. What is the area of an equilateral triangle with sides of length 10 ft?
12. A regular hexagon has sides of length 3 in. What is the area of the hexagon?
13. The area of an equilateral triangle is 363\begin{align*}36 \sqrt{3}\end{align*}. What is the length of a side?
14. 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 3 miles on this road, how much elevation have you gained in feet (5280 ft = 1 mile)?
15. 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 x 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 Triangle

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

45-45-90 Triangle

A 45-45-90 triangle is a special right triangle with angles of $45^\circ$, $45^\circ$, and $90^\circ$.