Discover one of the most famous theorems in mathematics: a² + b² = c²
Pythagoras' Theorem states that in a RIGHT-ANGLED triangle, the square of the hypotenuse (longest side) equals the sum of the squares of the other two sides.
a² + b² = c²
where c is the hypotenuse
Right-Angled Triangle Structure:
Pythagoras appears in EVERY GCSE maths paper - both Foundation and Higher. You MUST know this theorem!
Used in construction, navigation, engineering, architecture, and computer graphics. A truly practical theorem!
Understanding Pythagoras is essential before learning trigonometry (SOH CAH TOA) later in Year 9.
This theorem is named after the ancient Greek mathematician Pythagoras (c. 570-495 BC), although evidence suggests it was known to earlier civilizations. It's one of the most famous mathematical theorems in history!
Question: A right-angled triangle has sides a = 3cm and b = 4cm. Find the hypotenuse c.
Step 1: Write the formula: a² + b² = c²
Step 2: Substitute values: 3² + 4² = c²
Step 3: Calculate: 9 + 16 = c²
Step 4: Simplify: 25 = c²
Step 5: Square root: c = √25 = 5cm
Answer: c = 5cm
✓ Check: 5cm is longer than both 3cm and 4cm ✓
Question: Verify that a triangle with sides 3, 4, and 5 is right-angled.
Step 1: Check if a² + b² = c²
Step 2: Try 3² + 4² = 5²
Step 3: Calculate: 9 + 16 = 25
Step 4: Check: 25 = 25 ✓
Answer: YES, it is right-angled!
💡 The 3-4-5 triangle is the most famous Pythagorean triple!
Question: Find the hypotenuse when a = 5cm and b = 7cm. Round to 1 decimal place.
Step 1: a² + b² = c²
Step 2: 5² + 7² = c²
Step 3: 25 + 49 = c²
Step 4: 74 = c²
Step 5: c = √74 = 8.602...
Answer: c ≈ 8.6cm (1 d.p.)
Question: A ladder is placed 2m from a wall and reaches 6m up the wall. How long is the ladder?
Step 1: This forms a right-angled triangle
Step 2: a = 2m, b = 6m, find c (ladder)
Step 3: 2² + 6² = c²
Step 4: 4 + 36 = c²
Step 5: 40 = c²
Step 6: c = √40 = 6.32m
Answer: The ladder is 6.32m long
Question: A rectangle measures 8cm by 15cm. Calculate the length of its diagonal.
Step 1: The diagonal forms a right-angled triangle
Step 2: a = 8cm, b = 15cm
Step 3: 8² + 15² = c²
Step 4: 64 + 225 = c²
Step 5: 289 = c²
Step 6: c = √289 = 17cm
Answer: Diagonal = 17cm
Apply Pythagoras' Theorem to solve these problems:
What students often do wrong:
Students sometimes label the wrong side as 'c' (hypotenuse). The hypotenuse MUST be the longest side, opposite the right angle!
Correct approach: Always identify the right angle first (marked with □). The hypotenuse is ALWAYS the side opposite this angle and it's ALWAYS the longest side.
Memory tip: "Hypotenuse = Opposite the right angle = Longest side"
What students often do wrong:
Students calculate c² = 25 but forget to take the square root, giving answer c = 25 instead of c = 5.
Correct approach: Remember the formula gives you c² (c squared). To find c, you MUST take the square root: c = √(a² + b²)
Memory tip: "Square the sides, then square ROOT the answer!"
What students often do wrong:
Students try to use Pythagoras on triangles without a 90° angle. This theorem ONLY works for right-angled triangles!
Correct approach: ALWAYS check for the 90° angle (marked with □) before using Pythagoras. No right angle = use trigonometry instead (coming in later chapters).
In GCSE exams, always show your working: write the formula, substitute values, show calculations step-by-step. Even if your final answer is wrong, you can still earn method marks!
You've mastered Pythagoras' Theorem - a cornerstone of GCSE Mathematics!
Next: Finding the hypotenuse in more complex problems