Module 15 of 30Lesson 1 of 9 · T-C1 Pythagoras Theorem
Trigonometry strand · Unit T-C1 · Syllabus C6.1

Pythagoras’ Theorem

🔵 Core levela² + b² = c²

1 Learn it
🔑 In any right-angled triangle: a² + b² = c², where c is the HYPOTENUSE. Finding the hypotenuse → square, ADD, root. Finding a shorter side → square, SUBTRACT, root. The subtraction direction is where marks die.

Anchors: legs 6 and 8 → hyp = √(36 + 64) = 10. Hyp 13, one leg 5 → other = √(169 − 25) = 12.

Real-life disguises: the diagonal of a rectangle, a ladder against a wall, the distance between two coordinate points (CG-E2 was Pythagoras all along!). Answers may be surds when “exact” is demanded.

2 Watch it work
EXAMPLE 1

Legs 6 cm and 8 cm. Find the hypotenuse.

c = √(36 + 64) = √100 = 10 cm

EXAMPLE 2

Hypotenuse 13 cm, one leg 5 cm. Find the other leg.

√(169 − 25) = √144 = 12 cm — SUBTRACT when hunting a leg.

EXAMPLE 3

Find the diagonal of an 8 cm × 6 cm rectangle.

√(64 + 36) = 10 cm — the rectangle hides a right triangle.

EXAMPLE 4

A 5 m ladder leans against a wall, foot 1.4 m out. How high does it reach? 🔢

√(25 − 1.96) = √23.04 = 4.8 m — ladder = hypotenuse, so subtract.

3 Your turn — try first, then reveal
🚫 No calculator[2 marks]

Q1. Legs 9 cm and 12 cm. Find the hypotenuse.

👀 Show answer
✍️ How your working should look in the exam
√(81 + 144) = √225   ✓ M1  = 15 cm   ✓ A1
Why this works
Square, add, root — the 3-4-5 tripled.

🚫 No calculator[2 marks]

Q2. Hypotenuse 17 cm, one leg 8 cm. Find the other leg.

👀 Show answer
✍️ How your working should look in the exam
√(289 − 64) = √225   ✓ M1  = 15 cm   ✓ A1
Why this works
Hunting a leg → subtract. (The 8-15-17 triple.)

🔢 Calculator[3 marks]

Q3. A rectangle is 7.5 m by 4 m. Find its diagonal (3 s.f.).

👀 Show answer
✍️ How your working should look in the exam
d² = 56.25 + 16 = 72.25   ✓ M1 M1
d = 8.50 m   ✓ A1
Why this works
√72.25 = 8.5 exactly — the diagonal is always the hypotenuse.

🔢 Calculator[3 marks]

Q4. A 6 m ladder rests with its top 5.5 m up a wall. How far is its foot from the wall (3 s.f.)?

👀 Show answer
✍️ How your working should look in the exam
x² = 36 − 30.25 = 5.75   ✓ M1 M1
x = 2.40 m   ✓ A1
Why this works
The ladder is the hypotenuse — subtract to find the base run.

🚫 No calculator[2 marks]

Q5. Find the EXACT hypotenuse of a triangle with legs 2 cm and 6 cm.

👀 Show answer
✍️ How your working should look in the exam
√(4 + 36) = √40   ✓ M1  = 2√10 cm   ✓ A1
Why this works
“Exact” = simplified surd (N-E6). √40 = √4 × √10.

4 Common mistakes — don’t lose easy marks
  • Adding when the hypotenuse is GIVEN: hunting a leg means subtracting.
  • Forgetting the final root: 225 is c², not c.
  • Using Pythagoras without a right angle: it only works in right triangles (T-E4 handles the rest).
  • Rounding early in two-step problems.
5 How the exam asks this

Pythagoras earns direct marks and hides inside cones (M-C4), coordinate lengths (CG-E2) and 3-D work (T-E5). Next: SOHCAHTOA — when the angle joins the party.

✅ Take the Check Yourself quiz below, then continue to T-C2 SOHCAHTOA I.

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