Lesson 1: Pythagorean Theorem - Solving for the Hypotenuse

   Constructing Knowledge

Earlier in this lesson, the Pythagorean Theorem was introduced as a2 + b2 = c2, where the sides a and b are the legs of a right triangle, and side c is the hypotenuse. However, a right triangle's sides do not need to be labelled as a, b, and c for the formula to be used. Instead, the formula can be rewritten to reflect the given labelling of a triangle.

EXAMPLE 1

Rewrite the Pythagorean Theorem formula for the triangle shown.



Solution


Step 1: Label the sides of the triangle.



In this example, x is the hypotenuse so x must replace c in the formula a2 + b2 = c2. Sides y and z replace a and b.

The formula for this triangle is: y2 + z2 = x2 or z2 + y2 = x2.

y and z (the legs) are interchangable

   Multimedia

A video describing how to solve for an unknown hypotenuse in a right triangle is provided.


EXAMPLE 2

Determine the length of the hypotenuse in the triangle shown.



Solution


Step 1: Label the sides of the triangle.



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

\(\begin{align} a^{2}+b^{2}&=c^{2} \\ \\ 15^{2}+20^{2}&=c^{2} \\ \\ \left(15\times 15\right)+\left(20\times 20\right)&=c^{2} \\ \\ 225+400&=c^{2} \\ \\ \sqrt{625}&=\sqrt{c^{2}} \\ \\ 25&=c \end{align}\)

The hypotenuse is 25 cm long.


Now, it is your turn! Complete the questions in your Chapter 3, Lesson 1 Practice Makes Perfect that refer to Solving for the Hypotenuse.



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