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There is a rule connecting the side lengths of right-angled triangles. It is called Pythagoras' theorem, and it is true for all traingles with a right-angle.
Finding one side of a right-angled triangle when you know the other two
Look at the diagrams below. The areas of the squares are marked inside them. (The diagrams are drawn to different scales.) How are the squares' areas related to each other?
Did you spot it?
If you add the areas of the smallest two squares you get the area of the largest square.
In any right-angled triangle, the square of the longest side is the sum of the squares of the other two sides. This can be written in the formula:
a2 + b2 = c2
(where c is the longest side.)
That is Pythagoras' theorem.
Using Pythagoras theorem to solve problems
In your exam, you may be required to work out other mathematical problems using Pythagoras' theorem. This is easier than it sounds.
For instance, if you know two sides of a right-angled triangle, you can find the third like this:
- square (multiply by itself) the lengths you know
- add or subtract them
- find the square root
For these questions you will need to use the
button on your calculator.
- Question
Work out the missing measurements on the right-angled triangles below:
- Answer
Triangle 1
- Square the lengths you know: 62 = 36, 82 = 64.
- To find the square of the longest side, add them together 36 + 64 = 100.
Find the square root
= 10- ? = 10
Triangle 2
- Square the lengths you know 132 = 169, 52 = 25.
- You know the longest side, so subtract the shorter side from the longest 169 - 25 = 144.
Find the square root
= 12- ? = 12
Triangle 3
- Square the lengths you know 102 = 100, 72 = 49.
- To find the longest side, add them together 100 + 49 = 149.
Find the square root
= 12.206555
Give your answer to a sensible degree of accuracy, eg 2 decimal places (d.p.).
- ? = 12. 21 (2 d.p.).
Be careful to give your answer correct to the number of decimal places asked for. You may need to revise Rounding and estimating numbers.
- Question
Calculate the missing side lengths (shown with letters). Give your answers to one decimal place (1dp).
- Answer
Remember that if you use your calculator you should still show your working out. Writing out what you do will help you to see how it works, and show the examiner that you know the process.
Here is how to work out the answers:
- Triangle 1
- x2 = 212 + 202
- 212 + 202 = 841
- Find the square root of 841
- x = 29
- Triangle 2
- y2 = 3.22 + 5.52
- 3.22 + 5.52 = 40.49
- Find the square root of 40.49
- y = 6.4
- Triangle 3
- z2 = 192 - 162
- 192 - 162 = 105
- Find the square root of 105
- z = 10.2
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