IGCSE Maths / Pythagoras theorem
Pythagoras: finding a shorter side
Edexcel International GCSE Maths (4MA1), chapter 15: Pythagoras theorem, specification point 4.8a-f. Free lesson.
Read the revision notes · IGCSE Maths past papers
When a shorter side of a right-angled triangle is missing, subtract the square of the other shorter side from the square of the hypotenuse, then take the square root. Identifying the hypotenuse first helps you choose the right calculation and check your answer.
Decide whether to add or subtract
The hypotenuse is the side opposite the right angle, and it is the longest side. If the missing side is the hypotenuse, add the squares of the two shorter sides. If a shorter side is missing, the hypotenuse squared sits on its own, so subtract the other shorter side's square from it. Missing hypotenuse: c² = a² + b². Missing shorter side: a² = c² − b². A shorter side must come out shorter than the hypotenuse. Use this as a sense check.
Rearrange the rule to find a shorter side
Start with a² + b² = c². To get a² on its own, remove b² from the left. You must do the same to the right, because the two sides of an equation must stay equal. This gives the subtraction rule. For example, 15² − 8² = 225 − 64 = 161. At this point, a² is on its own, not a. Take the square root as a separate step at the end.
Find a missing shorter side
For a right-angled triangle with a hypotenuse of 13 cm and another shorter side of 5 cm, call the missing side x. Since 13 cm is the hypotenuse, subtract 5² from 13². Then take the square root to find x. Check the result by putting both shorter sides back into the addition rule. Their squares should add to the hypotenuse squared.
Avoid adding when you should subtract
If the hypotenuse is 13 cm and a shorter side is 5 cm, adding their squares gives 13² + 5² = 194. That cannot be right: 194 is greater than 13², so the supposed shorter side would be longer than the hypotenuse. The 13 cm side is the hypotenuse, so its square is the total and 5² must come off it. Use a² = c² − b², not c² + b², when a shorter side is missing. Subtract the smaller square from the bigger one. Reversing them gives a negative value, which cannot be the square of a length.
Apply the rule to a ladder
A 5 m ladder leans against a wall, with its foot 2 m from the wall. The ladder is the hypotenuse, so its square is the larger one. Subtract 2², then take the square root and round to 3 significant figures. Keep the unrounded value until the final answer. Rounding √21 early can affect the accuracy of later working. The check is 2² + 4.58² = 4 + 20.9764 = 24.9764, which is just under 25 because 4.58 m is rounded.
Remember the order
Find the hypotenuse before choosing the calculation. If a shorter side is missing, subtract first and take the square root last. Finally, check that the answer is shorter than the hypotenuse. Quick check: with a hypotenuse of 25 cm and one shorter side of 7 cm, the other shorter side is 24 cm.