While working in your Industrial Arts class one day, your Instructor asks you to use your 45-45-90 triangle to make a scale drawing. Unfortunately, you have two differently shaped triangles to use at your drafting table, and there aren't labels to tell you which triangle is the correct one to use.

You turn the triangles over and over in your hands, trying to figure out what to do, when you spot the ruler at your desk. Taking one of the triangles, you measure two of its sides. You determine that the first side is 7 inches long, and the second side is just a little under 9.9 inches. Can you determine if this is the correct triangle for your work?

### Special Triangle Ratios

Special right triangles are the basis of trigonometry. The angles \begin{align*}30^\circ, \ 45^\circ, \ 60^\circ\end{align*} and their multiples have special properties and significance in the unit circle (which you can read about in other Concepts). Students are usually required to memorize the ratios of sides in triangles with these angles because of their importance.

First, let’s compare the two ratios, so that we can better distinguish the difference between the two. For a \begin{align*}45-45-90\end{align*} triangle the ratio is \begin{align*}x:x:x\sqrt{2}\end{align*} and for a \begin{align*}30-60-90\end{align*} triangle the ratio is \begin{align*}x:x\sqrt{3}:2x\end{align*}. An easy way to tell the difference between these two ratios is the isosceles right triangle has two congruent sides, so its ratio has the \begin{align*}\sqrt{2}\end{align*}, whereas the \begin{align*}30-60-90\end{align*} angles are all divisible by 3, so that ratio includes the \begin{align*}\sqrt{3}\end{align*}. Also, if you are ever in doubt or forget the ratios, you can always use the Pythagorean Theorem. The ratios are considered a short cut.

#### Identifying Special Triangles

Determine if the set of lengths represents a special right triangle. If so, which one?

1. \begin{align*}8\sqrt{3}:24:16\sqrt{3}\end{align*}

Yes, this is a \begin{align*}30-60-90\end{align*} triangle. If the short leg is \begin{align*}x = 8\sqrt{3}\end{align*}, then the long leg is \begin{align*}8\sqrt{3} \cdot \sqrt{3} = 8 \cdot 3 = 24\end{align*} and the hypotenuse is \begin{align*}2 \cdot 8\sqrt{3} = 16\sqrt{3}\end{align*}.

2. \begin{align*}\sqrt{5}:\sqrt{5}:\sqrt{10}\end{align*}

Yes, this is a \begin{align*}45-45-90\end{align*} triangle. The two legs are equal and \begin{align*}\sqrt{5} \cdot \sqrt{2} = \sqrt{10}\end{align*}, which would be the length of the hypotenuse.

3. \begin{align*}6\sqrt{7}:6\sqrt{21}:12\end{align*}

No, this is not a special right triangle. The hypotenuse should be \begin{align*}12\sqrt{7}\end{align*} in order to be a \begin{align*}30-60-90\end{align*} triangle.

### Examples

#### Example 1

Earlier, you were asked if you have the right triangle for your work.

Since you know the ratios of lengths of sides for special triangles, you can test to see if the triangle in your hand is the correct one by testing the relationship:

hypotenuse = \begin{align*}\sqrt{2}x\end{align*}

where "x" is the length of the shorter sides. If you test this relationship with the triangle you are holding:

hypotenuse = \begin{align*}7\sqrt{2} = 9.87\end{align*} in

Yes, you are holding the correct triangle.

#### Example 2

Determine if the set of lengths below represents a special right triangle. If so, which one?

\begin{align*}3\sqrt{2}:3\sqrt{2}:6\end{align*}

The sides are the same length. This means that if the triangle is one of the special triangles at all, it must be a 45-45-90 triangle. To test this, we take either of the sides that are equal and multiply it by \begin{align*}\sqrt{2}\end{align*}:

\begin{align*}3\sqrt{2} \times \sqrt{2} = 3 \times \sqrt{4} = 3 \times 2 = 6\end{align*}

Yes, this triangle is a special triangle. It is a 45-45-90 triangle.

#### Example 3

Determine if the set of lengths below represents a special right triangle. If so, which one?

\begin{align*}4:2:2\sqrt{3}\end{align*}

It can immediately be seen that the second side is one half the length of the first side. This means that if it is a special triangle, it must be a 30-60-90 triangle. To see if it is indeed such a triangle, look at the relationship between the shorter side and the final side. The final side is \begin{align*}\sqrt{3}\end{align*} times the short side. So yes, this fulfills the criteria for a 30-60-90 triangle.

#### Example 4

Determine if the set of lengths below represents a special right triangle. If so, which one?

\begin{align*}13:84:85\end{align*}

It can be seen immediately that the lengths of sides given aren't a special triangle, since 84 is so close to 85. Therefore it can't be a 45-45-90 triangle, which would require \begin{align*}84\sqrt{2}\end{align*} to be a side or a 30-60-90 triangle, where a one of these two sides would have a relationship of multiplying/dividing by 2 or by \begin{align*}\sqrt{3}\end{align*}.

### Review

For each of the set of lengths below, determine whether or not they represent a special right triangle. If so, which one?

- \begin{align*}2:2:2\sqrt{2}\end{align*}
- \begin{align*}3:3:6\end{align*}
- \begin{align*}3:3\sqrt{3}:6\end{align*}
- \begin{align*}4\sqrt{2}:4\sqrt{2}:8\sqrt{2}\end{align*}
- \begin{align*}5\sqrt{2}:5\sqrt{2}:10\end{align*}
- \begin{align*}7:7\sqrt{2}:14\end{align*}
- \begin{align*}6\sqrt{5}:18\sqrt{5}:12\sqrt{5}\end{align*}
- \begin{align*}4\sqrt{6}:12\sqrt{2}:8\sqrt{6}\end{align*}
- \begin{align*}8\sqrt{15}:24\sqrt{5}:16\end{align*}
- \begin{align*}7\sqrt{6}:7\sqrt{6}:14\sqrt{3}\end{align*}
- \begin{align*}5\sqrt{7}:5\sqrt{14}:5\sqrt{7}\end{align*}
- \begin{align*}9\sqrt{6}:27\sqrt{2}:18\sqrt{6}\end{align*}

- Explain why if you cut any square in half along its diagonal you will create two 45-45-90 triangles.
- Explain how to create two 30-60-90 triangles from an equilateral triangle.
- Could a special right triangle ever have all three sides with integer lengths?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 1.7.