Pythagoras' theorem

What you will learn

identify the hypotenuse in a right-angled triangle,
apply Pythagoras’ theorem to find the hypotenuse when the two legs are given,
apply it in reverse to find a leg when the hypotenuse and one leg are known,
recognise common Pythagorean triples,
use it in real contexts - ladders, navigation, screen diagonals.

Worked example 0 Real-world example: is the gate square?

A carpenter builds a rectangular gate $1.2$ m wide and $0.9$ m tall. She measures the diagonal to check it is perfectly rectangular.

Expected diagonal: $d^2 = 1.2^2 + 0.9^2 = 1.44 + 0.81 = 2.25$ .
$d = \sqrt{2.25} = 1.5$ m.
She measures corner-to-corner: if her tape reads $1.5$ m, the gate is perfectly square. If not, the frame is skewed and needs adjusting.

Key idea: the $3$ - $4$ - $5$ triple (scaled to $0.9$ - $1.2$ - $1.5$ ) is used daily in construction to verify right angles without a protractor.

1. The theorem

In a right-angled triangle with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the right angle, and the longest):

A right-angled triangle: legs a and b meet at the right angle; c (the hypotenuse) is opposite the right angle.

Pythagoras' theorem

a^2 + b^2 = c^2.

Geometrically, the theorem says the square built on the hypotenuse has the same area as the two squares built on the legs, put together.

The classic picture: the big square on the hypotenuse equals the sum of the two smaller squares on the legs. Here 3² + 4² = 5², or 9 + 16 = 25.

2. Find the hypotenuse

Worked example E Very easy: 3-4-5 triangle

A right triangle has legs $3$ cm and $4$ cm. Find the hypotenuse.

c^2 = 3^2 + 4^2 = 9 + 16 = 25, \qquad c = \sqrt{25} = 5 \text{ cm}.

This is the most famous Pythagorean triple. Memorise it.

Worked example 1 Scaled triple: 6-8-10

Legs $6$ cm and $8$ cm. Find the hypotenuse.

c^2 = 6^2 + 8^2 = 36 + 64 = 100, \qquad c = \sqrt{100} = 10 \text{ cm}.

Notice this is the $3$ - $4$ - $5$ triple doubled.

Worked example 2 Irrational result

Legs $2$ and $3$ . Find the hypotenuse exactly and to 2 dp.

c^2 = 4 + 9 = 13, \qquad c = \sqrt{13} \approx 3.61.

3. Find a leg (reverse)

Worked example 3 Find a leg

Hypotenuse $13$ ; one leg $5$ . Find the other leg.

5^2 + b^2 = 13^2 \;\Longrightarrow\; b^2 = 169 - 25 = 144, \qquad b = 12.

Check: $5, 12, 13$ is a Pythagorean triple.

4. Common Pythagorean triples

These integer-sided right triangles are worth recognising:

(3,4,5), \quad (5,12,13), \quad (8,15,17), \quad (7,24,25), \quad (9,40,41), \quad (20,21,29).

Any multiple of a triple is also a triple: $(6, 8, 10)$ , $(9, 12, 15)$ , etc.

5. Real contexts

Worked example 4 The ladder problem

A $5$ m ladder leans against a wall with its foot $1.5$ m from the base of the wall. How high up the wall does it reach?

The ladder is the hypotenuse; the wall and the ground are the legs.

Let the height be $h$ . The ladder is the hypotenuse.

h^2 + 1.5^2 = 5^2 \;\Longrightarrow\; h^2 = 25 - 2.25 = 22.75, \qquad h \approx 4.77 \text{ m}.

Worked example 5 Screen diagonal

A TV is advertised as “40 inch” (the diagonal). If the screen is $35$ inches wide, how tall is it?

\text{height}^2 = 40^2 - 35^2 = 1600 - 1225 = 375, \qquad \text{height} \approx 19.4 \text{ in}.

Practice: Year 8 core

Fluency

Find the hypotenuse

Legs $3, 4$ .
Legs $5, 12$ .
Legs $6, 8$ .
Legs $9, 40$ .
Legs $2, 2$ . Give the exact value and the decimal.
Legs $1, \sqrt{3}$ . Give the exact hypotenuse.

Fluency

Find a leg

Hypotenuse $25$ , leg $7$ .
Hypotenuse $17$ , leg $8$ .
Hypotenuse $15$ , leg $9$ .
Hypotenuse $50$ , leg $30$ .
Hypotenuse $\sqrt{20}$ , leg $2$ . Give the exact value.

Fluency

Is it a right triangle?

A triangle with sides $a, b, c$ (largest $c$ ) is right-angled if and only if $a^2 + b^2 = c^2$ .

$6, 8, 10$ - right-angled?
$4, 5, 6$ - right-angled?
$7, 24, 25$ - right-angled?
$5, 12, 14$ - right-angled?

Reasoning

Explain and spot the mistake

Lee writes $5^2 + 12^2 = 17$ . Explain the error.
Explain why the hypotenuse must always be longer than either leg.
A triangle has sides $9, 12, 15$ . Is it right-angled? Is the triangle similar to any triple you recognise?
Sam says $7, 24, 26$ is a Pythagorean triple because the numbers “look similar” to $7, 24, 25$ . Is Sam right? Justify.

Problem solving

Real contexts

A ladder $6$ m long is placed with its foot $2$ m from a wall. How high does it reach?
A rectangular field is $30$ m by $40$ m. How far is the diagonal?
A guy wire supports a pole and is attached $8$ m from the foot of the pole, $15$ m up. How long is the wire?
A TV has a 50-inch diagonal and is $45$ inches wide. Is it taller than $20$ inches? Justify.
A ship sails $8$ km east and then $6$ km north. How far is it from its starting point?

Challenge

Reasoning

Harder triangles

A right-angled isosceles triangle has legs of length $a$ . Show that the hypotenuse is $a\sqrt{2}$ .
A rectangular prism measures $3$ cm by $4$ cm by $12$ cm. Find the length of the longest internal diagonal.
A rhombus has diagonals $10$ cm and $24$ cm. Find the side length.
An equilateral triangle has side $10$ cm. Find its height, to one decimal place.