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C6.1 Pythagoras’ theorem

Syllabus
0580–2028–2029
Topic
C6.1
Level
Core

Use Pythagoras' theorem

Pythagoras' theorem connects the three side lengths of a right-angled triangle. The hypotenuse is the side opposite the right angle and is always the longest side.

c^2=a^2+b^2

Label the hypotenuse cc before substituting. If cc is unknown, add the two shorter-side squares and take the positive square root. If a shorter side is unknown, subtract the known shorter-side square from c2c^2, then take the positive square root. Keep an exact surd when requested; otherwise round only the final length.

With shorter sides 77 cm and 2424 cm, c=72+242=625=25c=\sqrt{7^2+24^2}=\sqrt{625}=25 cm. If the hypotenuse is 1313 cm and one shorter side is 55 cm, the other is 132−52=144=12\sqrt{13^2-5^2}=\sqrt{144}=12 cm. The triples satisfy 72+242=2527^2+24^2=25^2 and 52+122=1325^2+12^2=13^2.

Use this theorem only when a right angle is known or can be proved. Never subtract when finding the hypotenuse. Conversely, if the square of the longest side equals the sum of the other two squares, the triangle is right-angled. Lengths use positive square roots and linear units.