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.
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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?
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.
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
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, $a^2 + b^2 \neq c^2$.
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.
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 |
Watch Me Solve It · 3 examples
- 1Identify $c$ (the largest)$c = 10$, $a = 6$, $b = 8$The largest number must be the hypotenuse if a right angle is possible.
- 2Compute $a^2 + b^2$$6^2 + 8^2 = 36 + 64 = 100$
- 3Compare with $c^2$$c^2 = 10^2 = 100$. Equal → Pythagorean triple.Notice 6-8-10 is just $2 \times$(3, 4, 5).
- 1Locate the right angle90° sits at vertex $Q$.
- 2Find the side opposite $Q$The side not touching $Q$ is $PR$.$PQ$ and $QR$ both touch $Q$ — they are the legs.
- 3Confirm with lengths$PR = 13$ is the longest. Verify: $5^2 + 12^2 = 169 = 13^2$ ✓
- 1Pick the largest as $c$$c = 15$, $a = 9$, $b = 12$
- 2Check $a^2 + b^2 = c^2$$9^2 + 12^2 = 81 + 144 = 225$ and $15^2 = 225$ ✓
- 3ConcludeEqual → yes, it IS a right-angled triangle.9, 12, 15 = $3 \times$(3, 4, 5).
Common Pitfalls
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
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
Quick Check · 5 questions
Show Your Working · 3 questions
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
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.
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.
Quick Check
1. C — The hypotenuse is opposite the right angle and is the longest side.
2. A — 3-4-5: $9 + 16 = 25 = 5^2$.
3. B — The theorem is $a^2 + b^2 = c^2$.
4. D — 10-24-26 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].
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 $a^2 + b^2 = c^2$.
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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