Finding a Shorter Side Using Pythagoras' Theorem
(KS3, Year 7)
In the formula, a and b are the lengths of the shorter sides and c is the length of the hypotenuse. The image below shows what we mean:
How to Find the Hypotenuse Using Pythagoras' Theorem
Question
What is the unknown length, a, of the right triangle below?Step-by-Step:
1
Start with the formula:
a = √(c2 − b2)
Don't forget: √ means square root and c2 = c × c (c squared), b2 = b × b.
2
Find the lengths of the sides from the right triangle.
In our example, the known side lengths are b = 6 and h = 10.
3
Substitute b = 6 and c = 10 into the formula.
a = √(102 − 62) = √((10 × 10) − (6 × 6))
c = √(100 − 36) = √64
a = 8
Answer:
The length of the shorter side, a, is 8.A Real Example of How to Find a Shorter Side Using Pythagoras' Theorem
In the example above, we have used the formula a = √(c2 − b2) to find the length of the side, a. This is just a rearrangement of the more memorable formula, a2 + b2 = c2 (see Note). If you find this simpler formula easier to remember, use it! Substitute in the lengths you know (replace the letters b and c with numbers) and then rearrange to find the side, a. There are two shorter sides, a and b. It does not really matter which of the shorter sides is labelled a and which is labelled b. The formula b = √(c2 − a2) can also be used.Interactive Widget
Here is an interactive widget to help you learn about Pythagoras' theorem.Rearranging Pythagoras' Theorem
Pythagoras' theorem states:Subtract b2 from both sides:
Take square roots of both sides:
This gives the formula for finding the length of the unknown shorter side when the hypotenuse and the other shorter side are known:
Top Tip
Leaving the Answer as a Square Root
To find the side, a, you have to find the square root of a2. Unless a2 is a square number, a will not be a whole number. It is sometimes best to leave the answer as a square root (also called a surd). For example, find the hypotenuse for the right triangle below, where b = 1 and c = 2:a2 + b2 = c2
a2 + 12 = 22
a2 = 22 − 12
a2 = 4 − 1
a2 = 3
Worksheet
This test is printable and sendable