Pythagoras theoremIB MYP Maths Extended: Revision notes

In a right-angled triangle the longest side, opposite the right angle, is the hypotenuse ($c$). Pythagoras theorem says $a^2+b^2=c^2,$ where $a$ and $b$ are the two shorter sides. The theorem only works for right-angled triangles. A Pythagorean triple is a set of whole numbers that fits, such as $3,4,5$ or $5,12,13$ or $8,15,17$.

To find the hypotenuse, add the squares of the two shorter sides and take the square root: $c=\sqrt{a^2+b^2}.$ Example: a rectangular field is $100$ m by $64$ m, so its diagonal is $\sqrt{100^2+64^2}=\sqrt{14096}=119$ m (3 s.f.). A quick check: the hypotenuse must be longer than either of the other two sides, but shorter than their sum.

To find a shorter side, subtract the square of the known shorter side from the square of the hypotenuse, then take the square root: $a=\sqrt{c^2-b^2}.$ Example: a $5$ m ladder has its foot $1.4$ m from a wall. It reaches $\sqrt{5^2-1.4^2}=\sqrt{23.04}=4.8$ m up the wall. Give answers to $3$ significant figures unless told otherwise, and keep the full calculator value until the final step.

A cuboid has a space diagonal from one corner to the opposite corner. Use Pythagoras twice, or the shortcut for a box $l\times w\times h$: $d=\sqrt{l^2+w^2+h^2}.$ Example: a box $12$ by $9$ by $8$. The base diagonal is $\sqrt{12^2+9^2}=15$, then the space diagonal is $\sqrt{15^2+8^2}=17$ cm. To see the right-angled triangle, draw it separately and label its three sides.

Pythagoras helps with ladders, ramps, distances across a field or room, cables and screen sizes.

1. Sketch the situation and mark the right angle.
2. Decide whether you need the hypotenuse or a shorter side.
3. Calculate, round sensibly and give units.
4. Answer in context, for example 'the ladder reaches $4.8$ m up the wall'. If the question asks for whole items (metres of cable sold in whole metres), round up to a sensible whole number.