Have you ever planned a flower garden where you needed to figure out the length of a diagonal? It is a special type of project, so take a look at this dilemma.

Ms. Kino’s class decided to do a community service project that everyone could enjoy. They decided to create a meditation garden that would be a rock garden.

Chas and Juanita took charge of the project. They drew a sketch of the rock garden and the rest of the class loved it so much that they instantly agreed to use the sketch that the pair had created. Here is their sketch.

“Let’s put a diagonal path in it,” Frankie suggested looking at the sketch.

“That’s a great idea, how long will the path be?” Chas asked.

The class wants to add a diagonal path. If they do that from one corner to another, how long will the path be?

**This Concept will teach you all that you need to know to solve this problem.**

### Guidance

There are a few types of right triangles it is particularly important to study. Their sides are always in the same ratio, and it is crucial to study the \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} and the \begin{align*}30^\circ-60^\circ-90^\circ\end{align*} triangles and understand the relationships between the sides. It will save you time and energy as you work through math problems both straight-forward and complicated.

Let’s start by learning about the \begin{align*}45^\circ-45^\circ-90^\circ\end{align*}.

**First, think about that \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} refers to. Those values refer to the angle measures in the right triangle. We can see that there is one 90 degree angle and that the other two angles have the same measure. This particular triangle is also** *isosceles***. An isosceles triangle has two side lengths that are the same. An isosceles right triangle will always have the same angle measurements: \begin{align*}45^\circ,45^\circ\end{align*}, and \begin{align*}90^\circ\end{align*} and will always have two side lengths that are the same. These characteristics make it a special right triangle.**

**Because these angles will always remain the same, the sides will always be in proportion. To find the relationship between the sides, use the Pythagorean Theorem.**

Take a look at this situation.

**The isosceles right triangle below has legs measuring 1 centimeter. Use the Pythagorean Theorem to find the length of the hypotenuse.**

**As the problem states, you can use the Pythagorean Theorem to find the length of the hypotenuse. Since the legs are 1 centimeter each, set both \begin{align*}a\end{align*} and \begin{align*}b\end{align*} equal to 1 and solve for \begin{align*}c\end{align*}.**

\begin{align*}a^2+b^2 &=c^2\\ (1)^2+(1)^2 &= c^2\\ 1+1 &= c^2\\ 2 &= c^2\\ \sqrt{2} &= \sqrt{c^2}\\ \sqrt{2} &= c\end{align*}

We can look at this and understand that there is also a 1 in front of the square root of two. This shows that the relationship between one side length and the length of the hypotenuse will always be the same. **The hypotenuse of an isosceles right triangle will always equal the product of one leg and \begin{align*}\sqrt{2}\end{align*}.**

*Write this down in your notebook under \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} special right triangles.*

Find each hypotenuse.

#### Example A

A triangle with side lengths of 9.

**Solution: \begin{align*}9 \sqrt{2}\end{align*}**

#### Example B

A triangle with side lengths of 15.

**Solution: \begin{align*}15 \sqrt{2}\end{align*}**

#### Example C

A triangle with side lengths of \begin{align*}3 \sqrt{2}\end{align*}

**Solution: \begin{align*}6\end{align*}**

Now let's go back to the dilemma from the beginning of the Concept.

**The first step in a word problem of this nature is to add important information to the drawing. Because the problem asks you to find the length of a path from one corner to another, you should draw that path in.**

**Once you draw the diagonal path, you can tell that this is a triangle question. Because both legs of the triangle have the same measurement (10 feet), this is an isosceles right triangle. The angles in an isosceles right triangle are \begin{align*}45^\circ,45^\circ\end{align*}, and \begin{align*}90^\circ\end{align*}.**

**In an isosceles right triangle, the hypotenuse is always equal to the product of the length of one leg and \begin{align*}\sqrt{2}\end{align*}. So, the length of the path will be the product of 10 and \begin{align*}\sqrt{2}\end{align*}, or \begin{align*}10 \sqrt{2} \ feet\end{align*}. This value is approximately equal to 14.14 feet.**

### Vocabulary

- Isosceles Triangle
- a triangle with two sides the same length.

- 45/45/90 Triangle
- a special right isosceles triangle.

### Guided Practice

Here is one for you to try on your own.

**What is the length of the hypotenuse in the triangle below?**

**Solution**

Since the length of the hypotenuse is the product of one leg and \begin{align*}\sqrt{2}\end{align*}, you can easily calculate this length. It is easy because we know that with any 45/45/90 degree triangle, that the hypotenuse is the product of one of the legs and the square root of 2.

One leg is 3 inches, so the hypotenuse will be \begin{align*}3 \sqrt{2}\end{align*}, or about 4.24 inches.

To get that answer, we took the square root of two on the calculator, 1.414 and then multiplied it times 3.

\begin{align*}3 \times 1.414 = 4.242\end{align*}

We rounded to get the answer.

### Video Review

### Practice

Directions: Find the missing hypotenuse in each \begin{align*}45^\circ-45^\circ-90^\circ\end{align*} triangle.

- Length of each leg = 5
- Length of each leg = 4
- Length of each leg = 6
- Length of each leg = 3
- Length of each leg = 7

Directions: Now use a calculator to figure out the approximate value of each hypotenuse. You may round to the nearest hundredth.

- \begin{align*}5 \sqrt{2}\end{align*}
- \begin{align*}4 \sqrt{2}\end{align*}
- \begin{align*}6 \sqrt{2}\end{align*}
- \begin{align*}3 \sqrt{2}\end{align*}
- \begin{align*}7 \sqrt{2}\end{align*}
- \begin{align*}8 \sqrt{2}\end{align*}
- \begin{align*}10 \sqrt{2}\end{align*}
- \begin{align*}13 \sqrt{2}\end{align*}
- \begin{align*}21\sqrt{2}\end{align*}
- \begin{align*}17 \sqrt{2}\end{align*}