Lesson 1 ~25 min Unit 3 · Measurement & Geometry +85 XP

Pythagoras' Theorem

Discover why $c^2 = a^2 + b^2$ unlocks every right-angled triangle.

Today's hook: Builders use a 3-4-5 triangle to make perfect right angles. Place a mark at 3 m on one wall and 4 m on another — if the diagonal is exactly 5 m, the corner is 90°. Why does this always work?

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Think First

warm-up

A right-angled triangle has two shorter sides of 3 cm and 4 cm. Without a ruler, can you predict what the longest side measures? Explain your reasoning.

Record your answer in your workbook.

The Big Idea

+5 XP

In a right-angled triangle, the square on the hypotenuse equals the sum of the squares on the other two sides.

The hypotenuse is always the longest side, sitting opposite the right angle. Label it $c$. The two shorter sides are the legs, labelled $a$ and $b$. Pythagoras proved that the areas of the squares built on $a$ and $b$ together equal the area of the square built on $c$.

$$c^2 = a^2 + b^2$$

Find $c$ first

Always identify the hypotenuse before substituting.

Square, then add

Square each leg, add the results, then take the square root.

Check with triples

3-4-5, 5-12-13 and 8-15-17 are quick-check families.

What You'll Master

objectives

Know

The statement of Pythagoras' Theorem: $c^2 = a^2 + b^2$
That the hypotenuse is always opposite the right angle
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17

Understand

Why squaring the sides works (square areas on each side)
The difference between the hypotenuse and the legs

Can Do

Calculate the hypotenuse given both legs
Identify Pythagorean triples and their multiples
Apply the theorem to real-world rectangles and diagonals

Words You Need

vocabulary

HypotenuseThe longest side of a right-angled triangle; always opposite the right angle.

Legs (shorter sides)The two sides that form the right angle; labelled $a$ and $b$.

Right angleAn angle of exactly 90°, marked with a small square in diagrams.

Square rootThe reverse of squaring: $\sqrt{25} = 5$ because $5^2 = 25$.

Pythagorean tripleThree whole numbers that satisfy $a^2 + b^2 = c^2$, e.g. 3, 4, 5.

TheoremA mathematical statement that has been proven to be always true.

Spot the Trap

heads-up

Wrong: $c = a + b$, so $c = 3 + 4 = 7$

Right: $c^2 = a^2 + b^2$, so $c = \sqrt{9+16} = \sqrt{25} = 5$

Wrong: Forgetting to take the square root at the end: $c = 25$

Right: $c^2 = 25$, so $c = \sqrt{25} = 5$ — always root at the end!

The Theorem Statement

+5 XP

Pythagoras' Theorem states: In any right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

Think of squares built on each side. The area of the big square (on $c$) equals the areas of the two smaller squares (on $a$ and $b$) added together. This is why we square every side — areas don't just add lengths, they add regions.

$$a^2 + b^2 = c^2$$

Identifying the Hypotenuse

+5 XP

The hypotenuse is always the longest side AND always sits directly opposite the right angle — regardless of how the triangle is oriented.

Three orientations — same rule: locate the right-angle box, then the side facing away from it is the hypotenuse. Never label a leg as $c$; the theorem only works when $c$ is the hypotenuse.

Hypotenuse = side opposite the right angle

Calculating $c$ — Step by Step

+5 XP

Four steps every time: Identify $a$ and $b$ → Square each → Add → Square root.

Label the hypotenuse $c$ (opposite right angle), shorter sides $a$ and $b$.
Write: $c^2 = a^2 + b^2$
Substitute: $c^2 = (\_)^2 + (\_)^2$
Calculate, then: $c = \sqrt{\_}$

On your calculator: type $a^2 + b^2$, press √.

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

Pythagorean Triples

+5 XP

A Pythagorean triple is a set of three whole numbers where $a^2 + b^2 = c^2$. Recognising these saves calculation time.

Core triples:

3 – 4 – 5 ($9 + 16 = 25$)
5 – 12 – 13 ($25 + 144 = 169$)
8 – 15 – 17 ($64 + 225 = 289$)

Multiples work too: multiply every side by the same number.
3-4-5 → 6-8-10 → 9-12-15 → 15-20-25

If sides are a triple: just recall — no calculator!

Common Pitfalls

heads-up

Adding sides instead of squaring

Writing $c = 3 + 4 = 7$ instead of squaring first.

Fix: Always write $c^2 = a^2 + b^2$ before substituting numbers.

Labelling the wrong side as hypotenuse

Picking the longest-looking side in the picture without checking it's opposite the right angle.

Fix: Find the right-angle box first, then the hypotenuse is directly across from it.

Stopping at $c^2$

Getting $c^2 = 25$ and writing "the hypotenuse is 25".

Fix: One more step — always take the square root: $c = \sqrt{25} = 5$.

Copy Into Your Books

Pythagoras' Theorem

$c^2 = a^2 + b^2$
$c$ = hypotenuse (longest side)
$a, b$ = legs (shorter sides)

Steps to find $c$

1. Identify $a$ and $b$
2. Square each: $a^2$, $b^2$
3. Add: $a^2 + b^2$
4. Square root: $c = \sqrt{a^2+b^2}$

Core Triples

3 – 4 – 5
5 – 12 – 13
8 – 15 – 17

Check

Is $c$ the longest? ✓
Is $c$ opposite the right angle? ✓
Did I take the square root? ✓

How are you completing this lesson?

Watch Me Solve It · 3 examples

Watch Me Solve It · Basic Triple

+15 XP per step

PROBLEM

A right-angled triangle has legs $a = 3$ cm and $b = 4$ cm. Find the hypotenuse $c$.

1
Write the formula
$c^2 = a^2 + b^2$
Start here every time — label hypotenuse as $c$.
2
Substitute and calculate
$c^2 = 3^2 + 4^2 = 9 + 16 = 25$
3
Take the square root
$c = \sqrt{25} = 5$ cm
Check: 3-4-5 is a Pythagorean triple — correct!

Answer$c = 5$ cm

Watch Me Solve It · 5-12-13 Triple

+15 XP per step

PROBLEM

A right-angled triangle has legs $a = 5$ cm and $b = 12$ cm. Find $c$.

1
Write the formula
$c^2 = a^2 + b^2$
2
Substitute
$c^2 = 5^2 + 12^2 = 25 + 144 = 169$
3
Square root
$c = \sqrt{169} = 13$ cm
5-12-13 is a core Pythagorean triple.

Answer$c = 13$ cm

Watch Me Solve It · Rectangle Diagonal

+15 XP per step

PROBLEM

A rectangle is 9 cm wide and 40 cm tall. Find the length of its diagonal.

1
Identify the right triangle
$a = 9$ cm, $b = 40$ cm, $c =$ diagonal
The diagonal of a rectangle cuts it into two right-angled triangles.
2
Apply the theorem
$c^2 = 9^2 + 40^2 = 81 + 1600 = 1681$
3
Square root
$c = \sqrt{1681} = 41$ cm
9-40-41 is also a Pythagorean triple!

AnswerDiagonal $= 41$ cm

Brain Trainer · 4 problems

Brain Trainer · Hypotenuse Drills

4 problems

Find the hypotenuse for each triangle. Work it out, then reveal the answer.

1 $a = 6$ cm, $b = 8$ cm. Find $c$.
$c^2 = 36 + 64 = 100$ → $c = 10$ cm
2 $a = 8$ cm, $b = 15$ cm. Find $c$.
$c^2 = 64 + 225 = 289$ → $c = 17$ cm
3 $a = 7$ cm, $b = 24$ cm. Find $c$.
$c^2 = 49 + 576 = 625$ → $c = 25$ cm
4 $a = 10$ cm, $b = 10$ cm. Find $c$ (to 1 decimal place).
$c^2 = 100 + 100 = 200$ → $c = \sqrt{200} \approx 14.1$ cm

Complete in your workbook.

Quick Check · 5 questions

A right triangle has legs 6 and 8. What is the hypotenuse?

+10 XP

Which side is the hypotenuse?

+10 XP

Which set is NOT a Pythagorean triple?

+10 XP

$a = 7$ cm, $b = 7$ cm. Find $c$ to the nearest cm.

+10 XP

A 5 m ladder leans against a wall with its base 3 m from the wall. How high does it reach?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. A rectangle is 24 cm long and 7 cm wide. Find the length of its diagonal. Show all working.

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. Show that 9, 12, 15 is a Pythagorean triple by checking $a^2 + b^2 = c^2$.

Answer in your workbook.

ReasonHard4 MARKS

Q8. A farm gate is 1.2 m wide and 0.9 m tall. (a) Find the length of the diagonal brace to 2 decimal places. (b) Find the area of the gate.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. A — $6^2 + 8^2 = 100$, so $c = 10$.

2. A — The hypotenuse is opposite the right angle.

3. C — $2^2 + 3^2 = 13 \neq 16 = 4^2$.

4. B — $c = \sqrt{98} \approx 9.9 \approx 10$ cm.

5. A — Height $= \sqrt{25-9} = 4$ m.

Model Answers

Q6 (3 marks): $c^2 = 24^2 + 7^2 = 576 + 49 = 625$. $c = \sqrt{625} = 25$ cm. (Note: 7-24-25 is a Pythagorean triple.)

Q7 (2 marks): $9^2 + 12^2 = 81 + 144 = 225 = 15^2$. Since $a^2 + b^2 = c^2$, the set 9, 12, 15 is a Pythagorean triple (a ×3 multiple of 3-4-5).

Q8 (4 marks): (a) $c^2 = 1.2^2 + 0.9^2 = 1.44 + 0.81 = 2.25$. $c = \sqrt{2.25} = 1.5$ m. (This is a 3-4-5 triple scaled by 0.3.) (b) Area $= 1.2 \times 0.9 = 1.08$ m².

Stretch Challenge · +25 XP, +10 coins

Rectangle Reverse Problem

A rectangle has a diagonal of 17 cm and one side of 8 cm. Find the other side, the perimeter, and the area of the rectangle. Show all working.

Reveal solution

Other side: $b = \sqrt{17^2 - 8^2} = \sqrt{289-64} = \sqrt{225} = 15$ cm. Perimeter $= 2(8+15) = 46$ cm. Area $= 8 \times 15 = 120$ cm².

Quick Review

$c^2 = a^2 + b^2$

Square both legs, add them, then square root to get the hypotenuse.

Hypotenuse = longest side

It is always the side sitting directly opposite the right angle.

Square then add

Never add $a + b$ first — square each side before adding.

Square root to finish

$c^2 = 25$ means $c = \sqrt{25} = 5$, not 25.

Right angle = 90°

Mark it with a small square in your diagrams.

Check with triples

3-4-5 and 5-12-13 give exact whole-number answers — recognise them!

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