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
Let’s start by learning about the
First, think about that
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
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
Write this down in your notebook under
Find each hypotenuse.
Example A
A triangle with side lengths of 9.
Solution:
Example B
A triangle with side lengths of 15.
Solution:
Example C
A triangle with side lengths of
Solution:
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
In an isosceles right triangle, the hypotenuse is always equal to the product of the length of one leg and
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
One leg is 3 inches, so the hypotenuse will be
To get that answer, we took the square root of two on the calculator, 1.414 and then multiplied it times 3.
We rounded to get the answer.
Video Review
Explore More
Directions:
Find the missing hypotenuse in each
 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.

52√ 
42√ 
62√ 
32√ 
72√ 
82√ 
102√ 
132√ 
212√ 
172√