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Lesson 6Finding Side Lengths of Triangles

Let’s find triangle side lengths.

Learning Targets:

  • I can explain what the Pythagorean Theorem says.

6.1 Which One Doesn’t Belong: Triangles

Which triangle doesn’t belong?

4 triangles are shown. Triangle B does not has a 90 degree angle.

6.2 A Table of Triangles

  1. Complete the tables for these three triangles:

    Three triangles on a square grid labeled “D,” “E,” and “F” with sides a, b, and c. The triangles have the following measurements: Triangle D: Horizontal side a is 2 units. Side b slants upward and to the left. Side c slants downward and to the right. The height of the triangle is 1. Triangle E: Horizontal side a is 2 units. Vertical side b is 1 unit. Diagonal side c slants downward and to the right and the triangle has a height of 1 unit. Triangle F: Horizontal side a is 2 units. Side b and side c are equal in length. The triangle has a height of 2 units.
    triangle a   b   c  
    D
    E
    F
    triangle a^2 b^2 c^2
    D
    E
    F
  2. What do you notice about the values in the table for Triangle E but not for Triangles D and F?
  3. Complete the tables for these three more triangles:
    Three triangles on a grid labeled “P,” “Q,” and “R” with sides a, b, and c. The triangles have the following measurements: Triangle P: Side a is 2 units. Side b slants upwards and to the left. Side c slants downward and to the right. The height of the triangle is 2. Triangle Q: Horizontal side a is 2 units. Vertical side b is 3 units. Diagonal side c slants downward and to the right and the triangle has a height of 3 units. Triangle R: Horizontal side a is 2 units. Side b and side c are equal in length. The triangle has a height of 3 units.
    triangle a   b   c  
    P
    Q
    R
    triangle a^2 b^2 c^2
    P
    Q
    R
  4. What do you notice about the values in the table for Triangle Q but not for Triangles P and R?
  5. What do Triangle E and Triangle Q have in common?

6.3 Meet the Pythagorean Theorem

  1. Find the missing side lengths. Be prepared to explain your reasoning.
  2. For which triangles does a^2+b^2=c^2 ?
Right triangle P is labeled with the side lengths of the square root of 8 and "b". The hypotenuse is labeled "c."

 
Right triangle Q has a hypotenuse labeled as "c." Side length a is the square root of 10 and side length b is the square root of 40.

 
Triangle R is shown on a graph with side lengths b as the square root of 17 and c as 5. Side length as is undetermined.

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 6 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=\sqrt5 , 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.

Glossary Terms

hypotenuse

The hypotenuse is the side of a right triangle that is opposite the right angle. It is the longest side of a right triangle. 

Here are some right triangles. Each hypotenuse is labeled.

a photo of 4 different sizes and shapes of right triangles. all sides are labeled "leg" or "hypotenuse"
legs

The legs of a right triangle are the sides that make the right angle.

Here are some right triangles. Each leg is labeled.

a photo of 4 different sizes and shapes of right triangles. all sides are labeled "leg" or "hypotenuse"
Pythagorean Theorem

The Pythagorean Theorem describes the relationship between the side lengths of right triangles.

The diagram shows a right triangle with squares built on each side.  If we add the areas of the two small squares, we get the area of the larger square.

The square of the hypotenuse is equal to the sum of the squares of the legs. This is written as a^2 + b^2=c^2 .

a right triangle with the length of 16, the height of 9, and the hypotenuse of 25

Lesson 6 Practice Problems

  1. Here is a diagram of an acute triangle and three squares.

    An acute triangle with squares along each side of the triangle. Each square has sides equal to the length of the side of the triangle it touches. The square on the bottom is touching the shortest side and is labeled 9. The square on the top right is touching the next longest side and is labeled 17. The square on the top left is touching the longest side and is unlabeled.

    Priya says the area of the large unmarked square is 26 square units because 9+17=26 . Do you agree? Explain your reasoning.

  2. m , p , and z represent the lengths of the three sides of this right triangle.

    A right triangle with the hypotenuse labeled as "m" and the side lengths labeled as "p" and "z"

    Select all the equations that represent the relationship between m , p , and z .

    1. m^2+p^2=z^2
    2. m^2=p^2+z^2
    3. m^2=z^2+p^2
    4. p^2+m^2=z^2
    5. z^2+p^2=m^2
    6. p^2+z^2=m^2

  3. The lengths of the three sides are given for several right triangles. For each, write an equation that expresses the relationship between the lengths of the three sides.

    1. 10, 6, 8
    2. \sqrt5, \sqrt3, \sqrt8
    3. 5, \sqrt5, \sqrt{30}
    4. 1, \sqrt{37} , 6
    5. 3, \sqrt{2}, \sqrt7

  4. Order the following expressions from least to greatest.

    25\div 10

    250,\!000 \div 1,\!000

    2.5 \div 1,\!000

    0.025\div 1

  5. Which is the best explanation for why \text-\sqrt{10} is irrational?

    1. \text- \sqrt{10} is irrational because it is not rational.

    2. \text- \sqrt{10} is irrational because it is less than zero.

    3. \text- \sqrt{10} is irrational because it is not a whole number.

    4. \text- \sqrt{10} is irrational because if I put  \text- \sqrt{10} into a calculator, I get -3.16227766, which does not make a repeating pattern.

  6. A teacher tells her students she is just over 1 and \frac{1}{2} billion seconds old.

    1. Write her age in seconds using scientific notation.
    2. What is a more reasonable unit of measurement for this situation?
    3. How old is she when you use a more reasonable unit of measurement?