E6.1 Pythagoras’ theorem

Learning objectives

E6.1.1Know and use Pythagoras’ theorem.• Know and use Pythagoras’ theorem.

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

c^2=a^2+b^2

Identify the hypotenuse $c$ before substituting. To find $c$ , add the squares of the two shorter sides and take the positive square root. To find a shorter side, subtract the other shorter-side square from $c^2$ , then take the positive square root. Keep an exact surd when required and round only the final length.

For shorter sides $9.6$ cm and $18$ cm, $c=\sqrt{9.6^2+18^2}=20.4$ cm. If the hypotenuse is $125$ m and one shorter side is $100$ m, the other is $\sqrt{125^2-100^2}=75$ m. The first calculation adds because the hypotenuse is unknown; the second subtracts because a shorter side is unknown.

Use the theorem only when a right angle is known or established. Never subtract when finding the hypotenuse, and do not forget the square root. Conversely, if the square of the longest side equals the sum of the other two squares, the triangle is right-angled. Lengths take positive roots and use linear units.

Next Topic

—E6.2 Right-angled trianglesRead