Lesson 1 ~25 min Unit 3 · Trigonometry +85 XP

Pythagoras' Theorem Review

Recall the rule that links the three sides of every right-angled triangle: $a^2 + b^2 = c^2$. Identify the hypotenuse, recognise Pythagorean triples, and see why the rule is always true.

Today's hook: Egyptian builders used a rope with 12 equally-spaced knots to make perfect right angles. They folded it into a triangle with sides 3, 4 and 5 knots. The angle between the sides of length 3 and 4 came out exactly 90°. How does Pythagoras' theorem explain this neat piece of ancient engineering?

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Think First

warm-up

A right-angled triangle has two short sides of length 3 cm and 4 cm. Without measuring, how long is the third (slanted) side? Try squaring the two short sides and adding them, what do you notice about the answer?

Record your answer in your workbook.

The Big Idea

+5 XP

Around 500 BCE, the Greek mathematician Pythagoras proved a stunning relationship: in any right-angled triangle, the square built on the longest side has the same area as the two squares built on the shorter sides put together.

Label the two shorter sides $a$ and $b$ (the legs) and the side opposite the right angle $c$ (the hypotenuse). Then $a^2 + b^2 = c^2$. The hypotenuse is always the longest side and is always opposite the right angle, not next to it. This is the only triangle rule where squaring the sides gives an exact equation.

$a^2 + b^2 = c^2$ where $c$ is the hypotenuse

Right angle only

Pythagoras' rule works only in right-angled triangles. Check the 90° marker first.

Hypotenuse = longest

$c$ is opposite the 90°. It is the longest side of the triangle, never one of the legs.

Square then add

Square BOTH legs, then add. The result equals the square of the hypotenuse.

What You'll Master

objectives

Know

Pythagoras' theorem: $a^2 + b^2 = c^2$
The hypotenuse is opposite the right angle and is the longest side
Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17

Understand

Why the squares on the two legs add to the square on the hypotenuse
How any multiple of a Pythagorean triple is also a Pythagorean triple
That the rule only works for right-angled triangles

Can Do

Label hypotenuse and legs on any right-angled triangle
Check whether three numbers form a Pythagorean triple
Apply $a^2 + b^2 = c^2$ in either direction

Words You Need

vocabulary

Right-angled triangleA triangle containing one 90° angle. The small square in the corner marks where the right angle is.

HypotenuseThe side opposite the right angle. Always the longest side of the right-angled triangle.

LegsThe two shorter sides that form the right angle. We usually call them $a$ and $b$.

Pythagorean tripleThree whole numbers $a$, $b$, $c$ where $a^2 + b^2 = c^2$. Examples: 3-4-5, 5-12-13.

SquaredMultiplied by itself. $7^2 = 7 \times 7 = 49$.

TheoremA mathematical statement that has been proven true. Pythagoras' theorem has many proofs.

Spot the Trap

heads-up

Wrong: “The hypotenuse is the slanted side.” Not always, the hypotenuse is the one OPPOSITE the right angle, no matter how the triangle is drawn.

Right: Locate the small square (right-angle mark) first. The side directly across from it is the hypotenuse.

Wrong: “$3 + 4 = 5$, so 3-4-5 is a Pythagorean triple.” That's just adding. You must check $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.

Right: Pythagoras only works for triangles with a 90° angle. For a 60°-60°-60° triangle, $a^2 + b^2 \neq c^2$.

Identifying the Hypotenuse

+5 XP

Before using the theorem you must locate the hypotenuse. Look for the small square marking the right angle, then trace across to the opposite side, that's $c$.

The right-angled corner has TWO sides touching it, these are the legs ($a$ and $b$). The remaining side, sitting OPPOSITE the right angle, is the hypotenuse ($c$). It does not matter how the triangle is rotated on the page: turn the triangle upside down and the hypotenuse is still the side across from the right angle.

Hypotenuse = side opposite the 90° (always longest)

Find the small square

The square symbol in the corner marks the 90°. The hypotenuse is directly across from it.

Longest side check

If a labelled side is shorter than another labelled side, it is NOT the hypotenuse.

Position doesn't matter

Rotating the triangle does not change which side is the hypotenuse.

Pythagorean Triples

+5 XP

A Pythagorean triple is a set of three whole numbers that exactly satisfies $a^2 + b^2 = c^2$. Recognising common triples lets you spot answers fast without a calculator.

Triple	Check	Doubled
3, 4, 5	$9+16=25$ ✓	6, 8, 10
5, 12, 13	$25+144=169$ ✓	10, 24, 26
8, 15, 17	$64+225=289$ ✓	16, 30, 34
7, 24, 25	$49+576=625$ ✓	14, 48, 50

Multiply ALL three numbers by the same factor → still a triple

Largest = hypotenuse

In 5-12-13, the 13 is always the hypotenuse. Largest number first into $c$.

Multiples count

9-12-15 is just 3$\times$(3, 4, 5), still a Pythagorean triple.

Memorise four

3-4-5, 5-12-13, 8-15-17, 7-24-25 cover most exam triangles.

Watch Me Solve It · 3 examples

Watch Me Solve It · Verify a triple

+15 XP per step

PROBLEM

Show that 6, 8, 10 form a Pythagorean triple.

1

Identify $c$ (the largest)

$c = 10$, $a = 6$, $b = 8$

The largest number must be the hypotenuse if a right angle is possible.
2

Compute $a^2 + b^2$

$6^2 + 8^2 = 36 + 64 = 100$
3

Compare with $c^2$

$c^2 = 10^2 = 100$. Equal → Pythagorean triple.

Notice 6-8-10 is just $2 \times$(3, 4, 5).

Answer$6^2 + 8^2 = 100 = 10^2$, yes, a Pythagorean triple.

Watch Me Solve It · Spot the hypotenuse

+15 XP per step

PROBLEM

A right-angled triangle has sides labelled $PQ = 5$ cm, $QR = 12$ cm and $PR = 13$ cm. The right angle is at $Q$. Which side is the hypotenuse?

1

Locate the right angle

90° sits at vertex $Q$.
2

Find the side opposite $Q$

The side not touching $Q$ is $PR$.

$PQ$ and $QR$ both touch $Q$, they are the legs.
3

Confirm with lengths

$PR = 13$ is the longest. Verify: $5^2 + 12^2 = 169 = 13^2$ ✓

Answer$PR = 13$ cm is the hypotenuse.

Watch Me Solve It · Test if a triangle is right-angled

+15 XP per step

PROBLEM

Are 9, 12, 15 the sides of a right-angled triangle?

1

Pick the largest as $c$

$c = 15$, $a = 9$, $b = 12$
2

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

$9^2 + 12^2 = 81 + 144 = 225$ and $15^2 = 225$ ✓
3

Conclude

Equal → yes, it IS a right-angled triangle.

9, 12, 15 = $3 \times$(3, 4, 5).

AnswerYes, $9^2 + 12^2 = 15^2 = 225$, so the triangle is right-angled.

Common Pitfalls

heads-up

Adding instead of squaring

Writing $3 + 4 = 5$ to "prove" 3-4-5 is a triple. That's not the rule.

Fix: Always SQUARE first, $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.

Using the rule on a non-right triangle

Pythagoras only works when there's a 90° angle. On any other triangle the rule fails.

Fix: Find the right-angle marker first. No square symbol = no Pythagoras.

Calling the slanted side the hypotenuse

If the triangle is drawn rotated, the slanted side might not be the hypotenuse.

Fix: The hypotenuse is opposite the right angle and is the longest side, check, don't assume.

Copy Into Your Books

The Theorem

$a^2 + b^2 = c^2$
$c$ = hypotenuse
Only in right-angled triangles

Hypotenuse

Opposite the 90°
Longest side
Always labelled $c$

Common triples

3, 4, 5
5, 12, 13
8, 15, 17
7, 24, 25

Multiples

6, 8, 10 = $2\times$(3,4,5)
9, 12, 15 = $3\times$(3,4,5)
Multiplying ALL three keeps it a triple

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Pythagoras Basics

4 problems

Four quick drills to lock in the theorem and the triples. Try each, then reveal the answer.

1 Which side of a right triangle is the hypotenuse?

It sits across from the right angle.The side opposite the 90°, also the longest
2 Is 5-12-13 a Pythagorean triple?

$5^2 + 12^2 = 25 + 144 = 169 = 13^2$.Yes, verified
3 Is 10-24-26 a Pythagorean triple?

$10^2 + 24^2 = 100 + 576 = 676 = 26^2$. It's $2\times$(5,12,13).Yes
4 Is 2-3-4 a Pythagorean triple?

$2^2 + 3^2 = 4 + 9 = 13$ but $4^2 = 16$. Not equal.No

Complete in your workbook.

Quick Check · 5 questions

Which side of a right-angled triangle is the hypotenuse?

+10 XP

Which set is a Pythagorean triple?

+10 XP

Pythagoras' theorem states:

+10 XP

Which set is also a Pythagorean triple? (Multiple of 5-12-13)

+10 XP

In a right-angled triangle with legs 9 and 12, which side is the hypotenuse?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. Test whether each set forms a Pythagorean triple. Show the check $a^2 + b^2$ vs $c^2$ in each case. (a) 8, 15, 17 (b) 6, 7, 9 (c) 9, 40, 41

Answer in your workbook.

UnderstandEasy2 MARKS

Q7. A right-angled triangle has its right angle at $B$. The three vertices are $A$, $B$, $C$ with sides $AB = 7$ cm, $BC = 24$ cm and $AC = 25$ cm. Name the hypotenuse and explain how you know.

Answer in your workbook.

ReasonHard4 MARKS

Q8. The Egyptians tied 12 evenly-spaced knots in a rope and folded it into a triangle. Explain mathematically why the triangle they made was guaranteed to have a right angle, and state where the right angle was located.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C The hypotenuse is opposite the right angle and is the longest side.

2. A3-4-5: $9 + 16 = 25 = 5^2$.

3. B The theorem is $a^2 + b^2 = c^2$.

4. D10-24-26 is $2\times$(5-12-13).

5. B The hypotenuse must be the longest side; the third side is 15 ($9^2+12^2=225=15^2$).

Show Your Working Model Answers

Q6 (3 marks): (a) $64 + 225 = 289 = 17^2$ ✓ triple [1]. (b) $36 + 49 = 85 \neq 81 = 9^2$, not a triple [1]. (c) $81 + 1600 = 1681 = 41^2$ ✓ triple [1].

Q7 (2 marks): The hypotenuse is $AC$ [1]. It is opposite the right angle at $B$, and it is the longest side at 25 cm. Check: $7^2 + 24^2 = 49 + 576 = 625 = 25^2$ [1].

Q8 (4 marks): 12 knots create 12 equal segments [1]. The simplest division into three sides is 3 + 4 + 5 segments [1]. Since $3^2 + 4^2 = 9 + 16 = 25 = 5^2$, by the converse of Pythagoras' theorem the triangle must be right-angled [1]. The right angle lies between the sides of length 3 and 4 (opposite the side of length 5) [1].

Stretch Challenge · +25 XP, +10 coins

The Visual Proof

Draw a square of side $a + b$. Inside, place four identical right-angled triangles (legs $a$ and $b$, hypotenuse $c$) around the edges. The shape left in the middle is a square of side $c$. By comparing two ways of writing the total area, can you derive $a^2 + b^2 = c^2$?

Reveal solution

Big square: $(a+b)^2 = a^2 + 2ab + b^2$. Also = $4 \cdot \tfrac{1}{2}ab + c^2 = 2ab + c^2$. Equate: $a^2 + 2ab + b^2 = 2ab + c^2$, so $a^2 + b^2 = c^2$.

Quick Review

The theorem

$a^2 + b^2 = c^2$ in any right-angled triangle

Hypotenuse

Opposite the right angle, always longest

Triples

3-4-5, 5-12-13, 8-15-17, 7-24-25

Multiples

$k\times$(triple) is still a triple

Square then add

Never just add, square first, then add

Right-angle only

No 90°? Then Pythagoras does not apply

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