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?

At the completion of this Concept, you'll be able to verify if this is the correct triangle to use.

### Watch This

James Sousa Solving Special Right Triangles

### Guidance

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.

#### Example A

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

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

**Solution:** 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*}.

#### Example B

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

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

**Solution:** 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.

#### Example C

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

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

**Solution:** 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.

### Vocabulary

**Special Triangle:** A **special triangle** is a triangle that has particular internal angles that cause the sides to have a certain length relationship with each other. Examples include a 45-45-90 triangle and a 30-60-90 triangle.

### Guided Practice

1. 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*}

2. 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*}

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

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

**Solutions:**

1. 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.

2. 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.

3. 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*}.

### Concept Problem Solution

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.

### Guided Practice

1. Find the length of the missing sides.

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

3. \begin{align*}x\end{align*} is the hypotenuse of a 45-45-90 triangle with leg lengths of \begin{align*}5\sqrt{3}\end{align*}.

**Answers:**

1. Use the \begin{align*}x : x : x \sqrt{2}\end{align*} ratio. \begin{align*}AB = 9 \sqrt{2}\end{align*} because it is equal to \begin{align*}AC\end{align*}. So, \begin{align*}BC = 9 \sqrt{2} \cdot \sqrt{2} = 9 \cdot 2 = 18\end{align*}.

2. Use the \begin{align*}x : x : x \sqrt{2}\end{align*} ratio. We need to solve for \begin{align*}x\end{align*} in the ratio.

\begin{align*}12 \sqrt{2} &= x \sqrt{2}\\ 12 &= x\end{align*}

3. \begin{align*}x=5\sqrt{3}\cdot \sqrt{2}=5\sqrt{6}\end{align*}

1. Find the length of the missing sides.

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

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

**Answers:**

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

2. We are given the hypotenuse.

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

The, the longer leg would be \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.

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

### Practice

- In a 30-60-90 triangle, if the shorter leg is 5, then the longer leg is __________ and the hypotenuse is ___________.
- In a 30-60-90 triangle, if the shorter leg is \begin{align*}x\end{align*}, then the longer leg is __________ and the hypotenuse is ___________.
- A rectangle has sides of length 6 and \begin{align*}6 \sqrt{3}\end{align*}. What is the length of the diagonal?
- 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.

- What is the height of an equilateral triangle with sides of length 3 in?
- What is the area of an equilateral triangle with sides of length 5 ft?
- 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.*) - The area of an equilateral triangle is \begin{align*}36 \sqrt{3}\end{align*}. What is the length of a side?
- If a road has a grade of \begin{align*}30^\circ\end{align*}, this means that its angle of elevation is \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)?

### Practice

- In an isosceles right triangle, if a leg is \begin{align*}x\end{align*}, then the hypotenuse is __________.
- In an isosceles right triangle, if the hypotenuse is \begin{align*}x\end{align*}, then each leg is __________.
- A square has sides of length 15. What is the length of the diagonal?
- A square’s diagonal is 22. What is the length of each side?
- A square has sides of length \begin{align*}6\sqrt{2}\end{align*}. What is the length of the diagonal?
- A square has sides of length \begin{align*}4 \sqrt{3}\end{align*}. What is the length of the diagonal?
- A baseball diamond is a square with 90 foot sides. What is the distance from home base to second base? (HINT: It’s the length of the diagonal).
- Four isosceles triangles are formed when both diagonals are drawn in a square. If the length of each side in the square is \begin{align*}s\end{align*}, what are the lengths of the legs of the isosceles triangles?

Find the lengths of the missing sides. Simplify all radicals.

### Practice

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?