A right-angled triangle has a hypotenuse of 29 cm and one shorter side of 20 cm. Find the third side.
- The side wanted is a shorter side, so subtract: a2 = 292 − 202.
- a2 = 841 − 400 = 441.
- a = √441 = 21.
Answer: 21 cm
Pythagoras’ theorem
Maths · Trigonometry. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
The idea
In a right-angled triangle the longest side is the hypotenuse. It is always the side opposite the right angle. Pythagoras’ theorem says that the square of the hypotenuse equals the sum of the squares of the other two sides.
To find the hypotenuse, square the two shorter sides, add, then take the square root. To find a shorter side, square the hypotenuse, subtract the square of the known side, then take the square root.
The theorem works only in right-angled triangles, so look for one hidden in the problem: the diagonal of a rectangle, the height of an isosceles triangle, a ladder against a wall, or a journey east then north. It also works backwards as a test for a right angle.
Rules to remember
- a2 + b2 = c2, where c is the hypotenuse
- Hypotenuse: c = √(a2 + b2)
- Shorter side: a = √(c2 − b2)
- Test: if the three sides fit a2 + b2 = c2, the triangle is right-angled, with the right angle opposite c
- Whole-number sets to know: 3, 4, 5; 5, 12, 13; 8, 15, 17; 7, 24, 25, and their multiples such as 6, 8, 10
- Check: the hypotenuse must be longer than each other side, and shorter than their sum
Common mistake
Students add the squares when they should subtract. If the side you want is not the hypotenuse, subtract, and check that your answer is shorter than the hypotenuse.
Worked example
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