Lesson 6Finding Side Lengths of Triangles

Learning Goal

Let’s find triangle side lengths.

Learning Targets

I can explain what the Pythagorean Theorem says.

Lesson Terms

hypotenuse
legs
Pythagorean Theorem

Warm Up: Which One Doesn’t Belong: Triangles

Which triangle doesn’t belong?

Activity 1: A Table of Triangles

Complete the tables for these three triangles:
triangle
$a$
$b$
$c$
D
E
F
triangle
$a^{2}$
$b^{2}$
$c^{2}$
D
E
F
What do you notice about the values in the table for Triangle E but not for Triangles D and F?
Complete the tables for these three more triangles:
triangle
$a$
$b$
$c$
P
Q
R
triangle
$a^{2}$
$b^{2}$
$c^{2}$
P
Q
R
What do you notice about the values in the table for Triangle Q but not for Triangles P and R?
What do Triangle E and Triangle Q have in common?

Activity 2: Meet the Pythagorean Theorem

Problem 1

Find the missing side lengths. Be prepared to explain your reasoning.

Find the missing side lengths. Be prepared to explain your reasoning.
Find the missing side lengths. Be prepared to explain your reasoning.

Problem 2

For which triangles does $a^{2} + b^{2} = c^{2}$ ?

Are you ready for more?

If the four shaded triangles in the figure are congruent right triangles, does the inner quadrilateral have to be a square? Explain how you know.

A square with side lengths of 14 units on a square grid. there is a second square inside the square. Each of the vertices of the inside square divides the side lengths of the large square into two lengths: 8 units and 6 units creating 4 right triangles.

Lesson Summary

A right triangle is a triangle with a right angle. In a right triangle, the side opposite the right angle is called the hypotenuse, and the two other sides are called its legs. Here are some right triangles with the hypotenuse and legs labeled:

Four right triangles of different sizes and orientations each with two legs and a hypotenuse opposite the right angle.

We often use the letters $a$ and $b$ to represent the lengths of the shorter sides of a triangle and $c$ to represent the length of the longest side of a right triangle. If the triangle is a right triangle, then $a$ and $b$ are used to represent the lengths of the legs, and $c$ is used to represent the length of the hypotenuse (since the hypotenuse is always the longest side of a right triangle). For example, in this right triangle, $a = \sqrt{20}$ , $b = \sqrt{5}$ , and $c = 5$ .

A right triangle with legs labeled “a” and “b.” The hypotenuse is labeled “c.”

Here are some right triangles:

Three right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 25” is attached to the hypotenuse. The triangle in the middle has the square labels “a squared equals 16” and “b squared equals 1” attached to each of the legs. The square labeled “c squared equals 17” is attached to the hypotenuse. The triangle on the right has the square labels “a squared equals 9” and “b squared equals 9” attached to each of the legs. The square labeled “c squared equals 18” is attached to the hypotenuse.

Notice that for these examples of right triangles, the square of the hypotenuse is equal to the sum of the squares of the legs. In the first right triangle in the diagram, $9 + 16 = 25$ , in the second, $1 + 16 = 17$ , and in the third, $9 + 9 = 18$ . Expressed another way, we have $a^{2} + b^{2} = c^{2}$ This is a property of all right triangles, not just these examples, and is often known as the Pythagorean Theorem. The name comes from a mathematician named Pythagoras who lived in ancient Greece around 2,500 BCE, but this property of right triangles was also discovered independently by mathematicians in other ancient cultures including Babylon, India, and China. In China, a name for the same relationship is the Shang Gao Theorem. In future lessons, you will learn some ways to explain why the Pythagorean Theorem is true for any right triangle.

It is important to note that this relationship does not hold for all triangles. Here are some triangles that are not right triangles, and notice that the lengths of their sides do not have the special relationship $a^{2} + b^{2} = c^{2}$ . That is, $16 + 10$ does not equal 18, and $2 + 10$ does not equal 16.

Two right triangles are indicated. A square is drawn using each side of the triangles. The triangle on the left has the square labels “a squared equals 16” aligned to the bottom horizontal leg and “b squared equals 10” aligned to the left leg. The square labeled “c squared equals 18 is aligned with the hypotenuse. The triangle on the right has the square labels of “a squared equals 10” aligned with the bottom leg and “b squared equals 2” aligned with the left leg. The square labeled “c squared equals 16” is aligned with the hypotenuse.

triangle	$a^{2}$	$b^{2}$	$c^{2}$
D
E
F

triangle	$a^{2}$	$b^{2}$	$c^{2}$
P
Q
R