Open Up Resources Logo

Lesson 8Finding Unknown Side Lengths

Let’s find missing side lengths of right triangles.

Learning Targets:

  • If I know the lengths of two sides, I can find the length of the third side in a right triangle.
  • When I have a right triangle, I can identify which side is the hypotenuse and which sides are the legs.

8.1 Which One Doesn’t Belong: Equations

Which one doesn’t belong?

3^2 + b^2 = 5^2  

b^2 = 5^2 - 3^2

3^2 + 5^2 = b^2

3^2 + 4^2 = 5^2

8.2 Which One Is the Hypotenuse?

Label all the hypotenuses with c .

6 triangles are shown. Triangle B is the only triangle without a right angle.

8.3 Find the Missing Side Lengths

  1. Find c .
    Right triangle P 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.
     
  2. Find b .
    Right triangle P has a hypotenuse of the square root of 26 and the side lengths of the square root of 8 and "b."
     
  3. A right triangle has sides of length 2.4 cm and 6.5 cm. What is the length of the hypotenuse?
  4. A right triangle has a side of length \frac14 and a hypotenuse of length \frac13 . What is the length of the other side?
  5. Find the value of x in the figure.
Two right triangles are shown sharing a side length. One has a hypotenuse of the square root of 34 and a length of 5. The other triangle has a hypotenuse of the square root of 18 and the side length is labeled "x."
 

Are you ready for more?

The spiral in the figure is made by starting with a right triangle with both legs measuring one unit each. Then a second right triangle is built with one leg measuring one unit, and the other leg being the hypotenuse of the first triangle. A third right triangle is built on the second triangle’s hypotenuse, again with the other leg measuring one unit, and so on.

The spiral in the figure is made by starting with a right triangle with both legs measuring one unit each. Then a second right triangle is built with one leg measuring one unit, and the other leg being the hypotenuse of the first triangle. A third right triangle is built on the second triangle’s hypotenuse, again with the other leg measuring one unit, and so on.
Find the length, x , of the hypotenuse of the last triangle constructed in the figure.

Lesson 8 Summary

There are many examples where the lengths of two legs of a right triangle are known and can be used to find the length of the hypotenuse with the Pythagorean Theorem. The Pythagorean Theorem can also be used if the length of the hypotenuse and one leg is known, and we want to find the length of the other leg. Here is a right triangle, where one leg has a length of 5 units, the hypotenuse has a length of 10 units, and the length of the other leg is represented by g . 

A right triangle, where one leg has a length of 5 units, the hypotenuse has a length of 10 units, and the length of the other leg is represented by the letter g.

Start with a^2+b^2=c^2 , make substitutions, and solve for the unknown value. Remember that c represents the hypotenuse: the side opposite the right angle. For this triangle, the hypotenuse is 10.

\begin{align} a^2+b^2&=c^2 \\ 5^2+g^2&=10^2 \\ g^2&=10^2-5^2 \\ g^2&=100-25 \\ g^2&=75 \\ g&=\sqrt{75} \\ \end{align}

Use estimation strategies to know that the length of the other leg is between 8 and 9 units, since 75 is between 64 and 81. A calculator with a square root function gives \sqrt{75} \approx 8.66 .

Lesson 8 Practice Problems

  1. Find the exact value of each variable that represents a side length in a right triangle.

    5 right triangles are shown all with a missing side length. The first triangles has a hypotenuse of 10 and a length of 8. H is missing. The second triangles has a hypotenuse of 6.5 and a length of 6. K is missing. The third triangles has a hypotenuse of 10 and a length of the square root of 10. N is missing. The forth triangles has a hypotenuse of 5 and a length of 2. M is missing. The fifth triangles has a hypotenuse of the square root of 85 and a length of the square root of 68. P is missing.

  2. A right triangle has side lengths of a , b , and c units. The longest side has a length of c units. Complete each equation to show three relations among a , b , and c .

    1. c^2=

    2. a^2=

    3. b^2=

  3. What is the exact length of each line segment? Explain or show your reasoning. (Each grid square represents 1 square unit.)

    1. A line segment labeled l on a square grid. One endpoint is 4 units directly down from the other endpoint.
    2. A line segment slanted upward and to the right, labeled m, on a square grid. The top endpoint is 2 units up and 4 units to the right from the bottom endpoint.
    3. A line segment labeled “q” on a square grid. The line segment starts at an intersection point on the grid and slants upward and to the right to an end point that is 4 units to the right and 5 units up.

  4. In 2015, there were roughly 1 \times 10^6 high school football players and 2 \times 10^3 professional football players in the United States. About how many times more high school football players are there? Explain how you know.

  5. Evaluate:

    1. \left(\frac{1}{2}\right)^3
    2. \left(\frac{1}{2}\right)^{\text-3}

  6. Here is a scatter plot of weight vs. age for different Dobermans. The model, represented by y = 2.45x + 1.22 , is graphed with the scatter plot. Here, x represents age in weeks, and y represents weight in pounds.

    a scatter plot showing a strong positive correlation between the age in weeks and the weight in pounds of puppies.
    1. What does the slope mean in this situation?
    2. Based on this model, how heavy would you expect a newborn Doberman to be?