Lesson 8

Finding Unknown Side Lengths

Let’s find missing side lengths of right triangles.

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 \(c\).

5 triangles labeled A, B, C, D, E. B is only triangle without right angle.

 

8.3: Find the Missing Side Lengths

  1. Find \(c\).
    Right triangle Q, a = square root 10, b = square root 40, hypotenuse = c.


     
  2. Find \(b\).
    Right triangle, P. legs = square root 8, b. hypotenuse = square root 26.

  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.

    2 Right triangles, share height side. on left, other leg = 5, hypotenuse = square root 34. on right, other leg = x, hypotenuse = square root 18.  


 



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.

Figure of a spiral 

Find the length, \(x\), of the hypotenuse of the last triangle constructed in the figure.

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 \(\sqrt{75} \approx 8.66\).

Video Summary

Glossary Entries

  • 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 squares built on each side
  • 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.

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

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

    Here are some right triangles. Each leg is labeled.

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