Pythagoras
Pythagoras' theorem links the three sides of a right-angled triangle: the square of the hypotenuse (the longest side, opposite the right angle) equals the sum of the squares of the other two sides, written a^2 + b^2 = c^2.
Method
- Identify the hypotenuse - the side opposite the right angle. It is always the longest side.
- If you need to find the hypotenuse, square the two shorter sides, add the results, then square root the total: c = sqrt(a^2 + b^2).
- If you need to find a shorter side, square the hypotenuse and the known shorter side, subtract, then square root: a = sqrt(c^2 - b^2).
- Always check your answer is sensible: the hypotenuse must be longer than either of the other two sides.
- For the distance between two coordinates, sketch a right-angled triangle using the horizontal and vertical differences as the two shorter sides.
Worked example
A right-angled triangle has shorter sides of 6 cm and 8 cm. Find the length of the hypotenuse.
- Square both shorter sides: 6^2 = 36 and 8^2 = 64.
- Add the results: 36 + 64 = 100.
- Square root the total: sqrt(100) = 10.
- The hypotenuse is 10 cm.
Practice questions
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Exam-style questions
Written in the style of a GCSE exam paper, with a full mark scheme.
A rectangle has length 24 cm and width 7 cm. Work out the length of its diagonal.
A ship sails 15 km due north, then 20 km due east. Work out the direct distance from the ship's starting point to its final position.
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