POP

E6.1 Pythagoras' Theorem

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01 — Explanation

What does Pythagoras' theorem actually say?

In any right-angled triangle, the hypotenuse (the longest side, opposite the right angle) is linked to the other two sides by:

c^2 = a^2 + b^2

where $c$ is the hypotenuse and $a$ , $b$ are the two shorter sides. The relationship only holds for right-angled triangles — check for the right-angle marker in the diagram before reaching for this formula. It works in both directions: given any two sides, the third can always be found, and the same equation can also be used to test whether a triangle is right-angled at all.

02 — Explanation

How do I find the hypotenuse?

Square both shorter sides, add them, then square root the total:

c = \sqrt{a^2+b^2}

For example, with legs $7$ cm and $9$ cm:

c = \sqrt{7^2+9^2} = \sqrt{49+81} = \sqrt{130} = 11.4 \text{ cm } (3\text{ s.f.})

03 — Explanation

How do I find one of the shorter sides instead?

If the hypotenuse is already known, rearrange first — subtract, don't add:

a = \sqrt{c^2-b^2}

Examiner Tip

The most common accuracy slip is adding when a side needs subtracting. Before substituting, write down which side is missing and whether it's the hypotenuse — if it is, you're solving for the largest value, so you add; if it isn't, you're solving for a smaller value, so you subtract from $c^2$ .

04 — Explanation

How do I check whether a triangle is right-angled in the first place?

This is the converse of Pythagoras' theorem: if the square of the longest side equals the sum of the squares of the other two, the triangle must contain a right angle — no diagram or angle marker needed.

Test $5$ , $12$ , $13$ : the longest side is $13$ , so check $5^2+12^2$ against $13^2$ .

5^2+12^2 = 25+144 = 169 = 13^2 \implies \text{right-angled}

Now test $5$ , $12$ , $14$ :

5^2+12^2 = 169, \qquad 14^2 = 196

Since $169 \ne 196$ , this triangle is not right-angled — no amount of rounding changes that conclusion.

Examiner Tip

Always square the longest side on its own first. Squaring the wrong side as if it were the hypotenuse is a common reason for wrongly concluding a triangle isn't right-angled when it actually is.

05 — Worked example

Worked Example 1: finding a height

A ladder $6.5$ m long leans against a vertical wall, with its foot $2.1$ m from the base of the wall. How high up the wall does the ladder reach?

Solution

The ladder is the hypotenuse ( $c = 6.5$ ), the ground distance is a shorter side ( $a = 2.1$ ), and the height $h$ up the wall is the other shorter side.

h^2 = c^2 - a^2 = 6.5^2 - 2.1^2 = 42.25 - 4.41 = 37.84

h = \sqrt{37.84} = 6.1514... \approx 6.15 \text{ m } (3\text{ s.f.})

Keep the unrounded value ( $37.84$ , not a rounded $h$ ) in the calculator until the final line — rounding too early is the easiest way to lose an accuracy mark in a multi-step problem.

06 — Worked example

Worked Example 2: an isosceles triangle

An isosceles triangle has two equal sides of length $13$ cm and a base of $10$ cm. Find the perpendicular height from the apex to the base.

Solution

Dropping a perpendicular from the apex to the base of an isosceles triangle always splits it into two identical right-angled triangles, each sharing the apex's height as one leg, half the base as the other, and the equal side as the hypotenuse.

\text{half base} = \frac{10}{2}=5 \text{ cm}

h = \sqrt{13^2-5^2} = \sqrt{169-25} = \sqrt{144} = 12 \text{ cm}

Spotting that an isosceles (or equilateral) triangle can be split along its line of symmetry into two right-angled triangles is one of the most useful habits in this topic — it turns a shape with no obvious right angle into a straightforward Pythagoras question.