Pythagoras' Theorem
FoundationIn a right-angled triangle, the squares of the two shorter sides (a, b) add up to the square of the hypotenuse (c): a² + b² = c². The key skill is spotting the hypotenuse — the longest side, always opposite the right angle.
- Pythagoras: a² + b² = c² (c is hypotenuse)
- Find hypotenuse: add then square root
- Find a leg: subtract then square root
1Summary 重點摘要
In a right-angled triangle, the squares of the two shorter sides (a, b) add up to the square of the hypotenuse (c): a² + b² = c². The key skill is spotting the hypotenuse — the longest side, always opposite the right angle.
2Vocabulary cards Vocabulary
The longest side of a right-angled triangle, always opposite the right angle.
A 90° angle, usually marked with a small square symbol.
A number multiplied by itself: 3² = 3 × 3 = 9.
The opposite of squaring: √9 = 3, because 3 × 3 = 9.
A triangle with one 90° angle.
The two shorter sides of a right-angled triangle (not the hypotenuse).
A math rule proven to be true. Pythagoras' theorem is one.
A fixed calculation rule — e.g. a² + b² = c².
3Step by step Steps
- 1. What is Pythagoras' theorem?Pythagoras' theorem is a rule about side lengths in a right-angled triangle.
A right-angled triangle has 3 sides: 2 short sides (called 'legs'), and 1 longest side (called the 'hypotenuse').
The rule: squares of the 2 short sides add up = square of the hypotenuse.
Formula: a² + b² = c²
(a, b are short sides, c is the hypotenuse) - 2. Step 1: Confirm it's right-angledBefore doing anything, confirm the triangle is right-angled.
How to tell? There's usually a small square symbol (□) at the right angle.
If no symbol, the question says 'right-angled triangle'.
Remember: if it's not right-angled, Pythagoras doesn't apply! - 3. Step 2: Label the hypotenuse cThe hypotenuse is the longest side, always opposite the right angle.
Example: a triangle with sides 3, 4, 5.
The longest is 5 → hypotenuse c = 5.
The other two sides 3 and 4 are the legs a and b.
Remember: c is ALWAYS the hypotenuse, don't mix up! - 4. Find hypotenuse: add then square rootExample: legs 3 and 4, hypotenuse = ?
Step 1: Add squares: 3² + 4² = 9 + 16 = 25.
Step 2: Square root: √25 = 5.
So hypotenuse = 5.
Formula: c = √(a² + b²) - 5. Find a leg: subtract then square rootExample: hypotenuse 13, one leg 5, other leg = ?
Formula rearranged: a² = c² − b².
Step 1: Subtract squares: 13² − 5² = 169 − 25 = 144.
Step 2: Square root: √144 = 12.
So other leg = 12.
Remember: finding a leg means subtract, not add! - 6. Check: is it right?Example: is 3, 4, 5 a right-angled triangle?
Check: 3² + 4² = 9 + 16 = 25.
5² = 25.
Both sides match → it is right-angled ✓.
Try again: 2, 3, 4.
2² + 3² = 4 + 9 = 13.
4² = 16.
13 ≠ 16 → not right-angled ✗.
4Diagrams Diagrams
5Worked example Example
A right-angled triangle has shorter sides 6 cm and 8 cm. Find the hypotenuse.
- Step 1: Label the sides: the hypotenuse c faces the right angle and is the longest.
- Step 2: Formula: c² = a² + b² = 6² + 8².
- Step 3: Square the numbers: 36 + 64 = 100, so c² = 100.
- Step 4: Square root: c = √100 = 10.
- Check: 6² + 8² = 100 = 10² ✓ (6-8-10 is the 3-4-5 triple doubled)
- Answer: Hypotenuse = 10 cm — to find the hypotenuse, add a² + b² then square-root.
6Common mistakes Mistakes
7Quick check Quick check
Q1Right-angled triangle legs 5 and 12, hypotenuse = ?
Q2Hypotenuse 13, one side 5, other side = ?
Q3Which is the hypotenuse of triangle 3, 4, 6? (if there is one)
Q4Right-angled triangle legs 6 and 8, hypotenuse = ?
Q5Hypotenuse 25, one side 7, other side = ?
8Recap Recap
- Pythagoras: a² + b² = c² (c is hypotenuse)
- Find hypotenuse: add then square root
- Find a leg: subtract then square root
- Only for RIGHT-angled triangles!
- Check: 3²+4²=5², if it matches, it's right