3.02 Finding the hypotenuse

Pythagoras' theorem states that 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 and b are the two shorter sides.

We can use the formula to find the length of the hypotenuse of any right-angled triangle, as long as we know the lengths of the other two sides.

In some cases, where the sides of the right-angled triangle form a Pythagorean triad, the exact length of the hypotenuse is an integer value. But for most cases we will end up with an irrational number, that is, a surd.

If we are asked to give an exact answer, or to answer as a surd, we can stop our working out when we arrive at a line of working such as c=\sqrt{11}, as this can not be simplified without losing some accuracy when it is rounded.

Examples

Example 1

Find the length of the unknown side c in the triangle below. Give the answer as a surd.

Worked Solution

Create a strategy

We can use the Pythagoras' theorem: a^{2}+b^{2} = c^{2}.

Apply the idea

\displaystyle c^{2}	\displaystyle =	\displaystyle a^{2}+b^{2}	Write the formula
\displaystyle c^{2}	\displaystyle =	\displaystyle 11^{2}+13^{2}	Substitute a and b
\displaystyle c^{2}	\displaystyle =	\displaystyle 121+169	Evaluate the squares
\displaystyle c^{2}	\displaystyle =	\displaystyle 290	Evaluate the sum
\displaystyle c	\displaystyle =	\displaystyle \sqrt{290}	Square root both sides

Reflect and check

The formula gives us a hypotenuse length of \sqrt{290} \approx 17.0294. This is larger than both our shorter sides, as we should expect. If we got a number smaller than one (or both) of the short sides we know we have made a mistake in our calculation.