7.04 Apply the Pythagorean theorem

The Pythagorean theorem states that in a right triangle the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. The theorem can be written algebraically:

a^2+b^2=c^2

where c represents the length of the hypotenuse and a \text{ and } b are the two shorter sides. To see why this is true check out the Investigation:  Discover the Pythagorean Theorem  .

We can use the formula to find any side if we know the lengths of the two others. Note that sometimes the equation is written in the reverse order: c^{2} = a^{2} + b^{2}.

If after we are done solving for the third side, we find that all three side lengths are whole numbers, the three side lengths may be referred to as a Pythagorean triple.

Examples

Example 1

Find the length of the hypotenuse, c in this triangle.

Worked Solution

Create a strategy

Use the Pythagorean theorem: c^2=a^2+b^2\,.

Apply the idea

Based on the given diagram, we are given a=8 and b=15.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
	\displaystyle =	\displaystyle 8^2+15^2	Substitute the values
	\displaystyle =	\displaystyle 64+ 225	Evaluate the squares
	\displaystyle =	\displaystyle 289	Evaluate
\displaystyle \sqrt{c^2}	\displaystyle =	\displaystyle \sqrt{289}	Apply the square root to both sides
\displaystyle c	\displaystyle =	\displaystyle 17\, \text{cm}	Evaluate the square root

Reflect and check

Because all the lengths for the sides of this triangle are whole numbers and they satisfy Pythagorean theorem, the numbers (8,15,17) form a Pythagorean triple.

If we need to find one of the shorter side lengths (a or b), using the formula we will have one extra step of rearranging to consider.

a^2 + b^2 = c^2

The value c is used to represent the hypotenuse which is the longest side of the triangle. The other two lengths are a, \,b. Sometimes diagrams or questions may use different variables, if no variables are given use a and b for either of the sides.

Examples

Example 2

Calculate the value of a in the triangle below.

Worked Solution

Create a strategy

Use the Pythagorean theorem: c^2=a^2+b^2\,

Apply the idea

We are given b=15 and c=17.

\displaystyle c^2	\displaystyle =	\displaystyle a^2+b^2	Use the Pythagorean theorem
\displaystyle 17^2	\displaystyle =	\displaystyle a^2+15^2	Substitute the values
\displaystyle 17^2 -15^2	\displaystyle =	\displaystyle a^2	Subtract 15^2 from both sides of the equation
\displaystyle 289 - 225	\displaystyle =	\displaystyle a^2	Evaluate the squares
\displaystyle 64	\displaystyle =	\displaystyle a^2	Evaluate the subtraction
\displaystyle \sqrt{64}	\displaystyle =	\displaystyle \sqrt{a^2}	Apply the square root to both sides
\displaystyle 8	\displaystyle =	\displaystyle a	Evaluate the square root

Watch question walkthrough

A Pythagorean triple (sometimes called a Pythagorean triad) is an ordered triple (a,b,c) of three positive integers so that a^2+b^2=c^2.

If (a,b,c) is a triple then (b,a,c) is also a triple, since b^2+a^2 is the same as a^2+b^2. So the order of the first two numbers in the triple doesn't matter.

(6,8,10) is also a Pythagorean triple, but it can be considered a multiple of another known Pythagorean triple, since 6, \,8 and 10 have a common factor of 2. If we divide each number in the triple by this common factor, we get the known Pythagorean triple (3,4,5).

A triangle whose sides form a Pythagorean triple will always be a right triangle.

Examples