Solving for the Length of a Leg of a Right Triangle

Lesson 1: Pythagorean Theorem - Solving for the Length of a Leg of a Right Triangle

   Constructing Knowledge

There are times where you calculate for one of the legs instead of the hypotenuse. You will initially follow the same steps (labelling the triangle, writing a formula) however, after the substitution the steps to solve become slightly different.

   Multimedia

A video describing how to solve for an unknown leg of a right triangle is provided.

EXAMPLE 1

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Solve for the missing side length of the triangle shown.

Solution

Step 1: Label the sides of the triangle.



Step 2: Write the formula, then substitute values and solve by subtracting.

\(\begin{align} a^{2}+b^{2}&=c^{2} \\ \\ 12^{2}+b^{2}&=17^{2} \\ \\ 144+b^{2}&=289 \\ \\ 144-{\color{red}{144}}+b^{2}&=289-{\color{red}{144}} \\ \\ b^{2}&=145 \\ \\ \sqrt{b^{2}}&=\sqrt{145} \\ \\ b&=12.04 \\ \end{align}\)

The length of side b is approximately 12.0 units.

EXAMPLE 2

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What is the length of the rectangle shown?

Solution

The sides of the right triangle are already labelled. Write the formula, substitute known values, and solve.

\(\begin{align} l^{2}+w^{2}&=d^{2} \\ \\ l^{2}+9^{2}&=13^{2} \\ \\ l^{2}+81&=169 \\ \\ l^{2}+\cancel{81-{\color{red}{81}}}&=169-{\color{red}{81}} \\ \\ l^{2}&=88 \\ \\ \sqrt{l^{2}}&=\sqrt{88} \\ \\ l&=9.38 \\ \end{align}\)

The length of the rectangle is approximately 9.4 cm.




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