Year 9: Pythagoras’ theorem

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Pythagoras’ Theorem is the backbone of right-angled triangle geometry. Whenever you have a right-angled triangle, you can use this famous rule to work out the lengths of its sides. From basic measurement tasks to more advanced trigonometry, you’ll be using Pythagoras well into senior maths.

It is named after the ancient Greek mathematician Pythagoras. It’s a fundamental tool for geometry, trigonometry, and practical problems like measuring distances.

1. What is Pythagoras’ Theorem?

Pythagoras’ Theorem states: for any right-angled triangle with sides $a$, $b$, and the longest side (the hypotenuse) $c$, the following relationship always holds:

$a\mathrm{²}+b\mathrm{²}=c\mathrm{²}$

This only applies to right-angled triangles!

In order words:

hypotenuse² = (leg₁)² + (leg₂)²

2. Identifying the sides correctly

In a right-angled triangle:

• The hypotenuse is always the side opposite the right angle (and it’s the longest side).
• The other two sides (often labelled $a$ and $b$) are the legs or “short sides.”

Make sure you know which side is the hypotenuse before you plug values into the theorem!

3. Finding the hypotenuse

When you know both “short sides” ($a$ and $b$), you can calculate the hypotenuse $c$ by:

$c=\surd (a\mathrm{²}+b\mathrm{²})$

Example: If you have a triangle with legs $a=6\text{cm}$ and $b=8\text{cm}$, then

$c=\surd (6\mathrm{²}+8\mathrm{²})=\surd (36+64)=\surd 100=10\text{cm}$

4. Finding a shorter side

When the hypotenuse $c$ is known and one short side $a$ is known, rearrange Pythagoras to find the other short side $b$:

$b=\surd (c\mathrm{²}–a\mathrm{²})$

Example: If $c=13\text{cm}$ and $a=5\text{cm}$,

$b=\surd (13\mathrm{²}–5\mathrm{²})=\surd (169–25)=\surd 144=12\text{cm}$

5. Pythagorean triplets

You might notice some right-angled triangles use “nice” whole-number sides, called Pythagorean triplets (like $3,4,5$ or $6,8,10$). These come from $a\mathrm{²}+b\mathrm{²}=c\mathrm{²}$ neatly matching perfect squares. It’s good to memorise the smaller ones—they pop up often in geometry tasks.

6. Checking If a Triangle Is Right-Angled

If you know all three sides, you can check if it's a right-angled triangle by testing if:

a² + b² = c²

Example: Sides 6 cm, 8 cm, and 10 cm
6² + 8² = 36 + 64 = 100 = 10² → It's a right-angled triangle

7. Real-world applications

• Mapping and navigation: Finding diagonal distances between two points.
• Carpentry: Checking if corners are perfectly square (e.g., measuring building frames).
• Computer graphics: Calculating distances between pixels on a screen.

8. Common mistakes

• Forgetting which side is the hypotenuse: Always the side opposite the right angle.
• Using the theorem in non-right triangles: It only works with a 90° angle.
• Mixing up addition/subtraction when finding a short side: Use $c\mathrm{²}–a\mathrm{²}$, not $a\mathrm{²}–c\mathrm{²}$.
• Not simplifying the square root properly: Check if perfect squares lurk under the root!