13.03 Finding a short side

We have been using Pythagoras' theorem to relate the three sides of a right-angled triangle together.

Up until now we have been using this formula to find the length of the hypotenuse, knowing the length of the two short sides.

We can use the same formula to find the length of a short side, knowing the length of the hypotenuse and the length of the other short side. The only difference when finding a short side is that we can put the numbers in the wrong way around if we aren't careful.

The most important thing to remember when finding a short side is that the two lengths need to go into different parts of the formula.

If you get the lengths around the wrong way, you will probably end up with the square root of a negative number (and a calculator error).

We can also rearrange the equation before we perform the substitution, to find formulas for each side length.

To find a shorter side: a^{2}=c^{2}-b^{2} or b^{2}=c^{2}-a^{2}. We can take the square root of both sides to give us the following formulas: a=\sqrt{c^{2}-b^{2}} or b=\sqrt{c^{2}-a^{2}}.

Examples

Example 1

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

Worked Solution

Create a strategy

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

Apply the idea

\displaystyle s^{2}	\displaystyle =	\displaystyle 16^{2}-7^{2}	Substitute a,\,b,\, and c
\displaystyle s^{2}	\displaystyle =	\displaystyle 256-49	Evaluate the squares
\displaystyle s^{2}	\displaystyle =	\displaystyle 207	Evaluate the subtraction
\displaystyle s	\displaystyle =	\displaystyle \sqrt{207}	Take the square root of both sides