Mathematics • Year 9 • Unit 3 • Lesson 1
Pythagoras' Theorem Review
Build fluency with $a^2 + b^2 = c^2$ in any right-angled triangle: identify the hypotenuse, check whether three numbers form a Pythagorean triple, and recognise common triples (3-4-5, 5-12-13, 8-15-17, 7-24-25) and their multiples.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Show that 6, 8, 10 form a Pythagorean triple.
Step 1 — Identify the hypotenuse $c$.
$c = 10$, $a = 6$, $b = 8$.
Reason: the hypotenuse must be the LARGEST of the three numbers — 10 is the largest.
Step 2 — Compute $a^2 + b^2$.
$6^2 + 8^2 = 36 + 64 = 100$.
Reason: square BOTH legs, then add — don't just add 6 + 8.
Step 3 — Compute $c^2$ and compare.
$c^2 = 10^2 = 100$. So $a^2 + b^2 = c^2$ ✓
Reason: equal means the rule holds — by the converse of Pythagoras' theorem, the triangle is right-angled.
Step 4 — Spot the pattern (bonus).
6, 8, 10 $= 2 \times$ (3, 4, 5).
Reason: any multiple of a triple is still a triple. Spotting this saves time in exams.
Answer: Yes — $\mathbf{6^2 + 8^2 = 100 = 10^2}$, so 6-8-10 is a Pythagorean triple.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Show that 9, 12, 15 form a Pythagorean triple.
Step 1 — Identify the hypotenuse: $c = $ __________ (the __________ of the three numbers).
Step 2 — Compute $a^2 + b^2$:
$9^2 + 12^2 = \_\_\_\_ + \_\_\_\_ = \_\_\_\_\_$
Step 3 — Compute $c^2$ and compare:
$c^2 = \_\_\_^2 = \_\_\_\_\_$. So $a^2 + b^2 \_\_\_ c^2$ (equal / not equal).
Step 4 — Spot the pattern:
9, 12, 15 = __ $\times$ (3, 4, 5).
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (one quick check). The middle two are standard (slightly harder numbers). The last two are extension (reasoning, not just checking).
Foundation — quick checks
3.1 Test whether 5, 12, 13 is a Pythagorean triple. Show the squares. 1 mark
3.2 Test whether 8, 15, 17 is a Pythagorean triple. Show the squares. 1 mark
3.3 A right-angled triangle has legs $a = 3$ cm and $b = 4$ cm. What is the hypotenuse $c$? You may use the 3-4-5 triple. 1 mark
3.4 In a right-angled triangle the right angle is at vertex $B$ and the three sides are $AB = 7$, $BC = 24$, $AC = 25$. Which side is the hypotenuse? Briefly explain. 1 mark
Standard — slightly harder numbers
3.5 Test whether 9, 40, 41 is a Pythagorean triple. Show the squares. 2 marks
3.6 Test whether 6, 7, 9 is a Pythagorean triple. If not, write one short sentence saying how you know. 2 marks
Extension — push your thinking
3.7 The triple 5-12-13 is given. Without checking each one, write down THREE other Pythagorean triples obtained by multiplying every term by the same whole number. 3 marks
3.8 A classmate says “3 + 4 = 5, so 3-4-5 is a Pythagorean triple.” In one or two sentences explain (i) what they have done wrong, and (ii) the correct check. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded 9, 12, 15)
Step 1: $c = \mathbf{15}$ (the largest of the three numbers).
Step 2: $9^2 + 12^2 = \mathbf{81} + \mathbf{144} = \mathbf{225}$.
Step 3: $c^2 = \mathbf{15}^2 = \mathbf{225}$. So $a^2 + b^2 \mathbf{=} c^2$ ✓.
Step 4: 9, 12, 15 = $\mathbf{3} \times$(3, 4, 5).
3.1 — Is 5, 12, 13 a triple?
$5^2 + 12^2 = 25 + 144 = 169$ and $13^2 = 169$. Equal — $\mathbf{yes}$, it is a Pythagorean triple.
3.2 — Is 8, 15, 17 a triple?
$8^2 + 15^2 = 64 + 225 = 289$ and $17^2 = 289$. Equal — $\mathbf{yes}$, it is a Pythagorean triple.
3.3 — Legs 3 and 4, find $c$
This is the 3-4-5 triple. $c = \mathbf{5}$ cm. (Check: $3^2 + 4^2 = 9 + 16 = 25 = 5^2$.)
3.4 — Hypotenuse of triangle $ABC$
The hypotenuse is $\mathbf{AC}$. The right angle is at $B$, and the hypotenuse is always the side OPPOSITE the right angle. It is also the longest at 25. Check: $7^2 + 24^2 = 49 + 576 = 625 = 25^2$ ✓.
3.5 — Is 9, 40, 41 a triple?
$9^2 + 40^2 = 81 + 1600 = 1681$ and $41^2 = 1681$. Equal — $\mathbf{yes}$, it is a Pythagorean triple.
3.6 — Is 6, 7, 9 a triple?
$6^2 + 7^2 = 36 + 49 = 85$ and $9^2 = 81$. $85 \ne 81$, so $\mathbf{no}$ — 6, 7, 9 is NOT a Pythagorean triple.
How I know: $a^2 + b^2$ does not equal $c^2$, so the converse of Pythagoras' theorem fails.
3.7 — Three multiples of 5-12-13
Any whole-number multiple works. Examples:
$2 \times$(5, 12, 13) $= \mathbf{10, 24, 26}$.
$3 \times$(5, 12, 13) $= \mathbf{15, 36, 39}$.
$4 \times$(5, 12, 13) $= \mathbf{20, 48, 52}$.
Reason: multiplying every side by $k$ scales the squares by $k^2$, so $a^2 + b^2 = c^2$ still holds.
3.8 — The classmate's mistake
(i) The classmate just added the numbers ($3 + 4 = 7$, not 5 — and even getting 5 by accident isn't a valid proof). Pythagoras' rule is about squares, not raw sums.
(ii) The correct check is $3^2 + 4^2 = 9 + 16 = 25$, and $5^2 = 25$. Because the squares are equal, 3-4-5 IS a Pythagorean triple — but only the squaring check proves it.