Pythagoras’ Theorem Revision Notes for AQA GCSE Maths

Pythagoras' theorem

Understanding the basics

Pythagoras' theorem might sound complicated, but it's actually quite straightforward once you understand the fundamentals. This powerful mathematical tool helps us find unknown side lengths in right-angled triangles, and it's essential for solving many geometry problems.

The theorem states that in any right-angled triangle, the square of the hypotenuse (the longest side) equals the sum of the squares of the other two sides. This relationship is expressed through the famous formula:

$a^2 + b^2 = c^2$

Where:

$a$ and $b$ are the two shorter sides
$c$ is the hypotenuse (longest side, opposite the right angle)

infoNote

The hypotenuse is always the longest side in a right-angled triangle and is positioned directly opposite the 90° angle. This is crucial for correctly applying the formula.

Key rules to remember

There are several important points you must keep in mind when using Pythagoras' theorem:

Only works for right-angled triangles: This theorem can only be applied to triangles that contain a 90° angle. If your triangle doesn't have a right angle, you cannot use this method.

Uses two sides to find the third: You need to know the lengths of two sides to calculate the third side using this theorem.

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Critical Rule: The hypotenuse is always the longest side and is positioned opposite the right angle. Getting this placement wrong is a common source of errors.

Always check your answer makes sense: After calculating, ensure your result is reasonable - the hypotenuse should be longer than either of the other two sides.

Finding exact lengths

When questions ask for an exact length, you should express your answer as a simplified surd rather than a decimal approximation. Let's look at how this works in practice.

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Worked Example: Finding Exact Length

Given a right-angled triangle with sides AB = 6 metres and AC = 3 metres, find BC.

Step 1: Start with the basic formula: $a^2 + b^2 = c^2$

Step 2: Substitute the known values: $3^2 + BC^2 = 6^2$

Step 3: Rearrange to solve for $BC^2$ : $BC^2 = 36 - 9 = 27$

Step 4: Take the square root: $BC = \sqrt{27} = 3\sqrt{3}$ metres

Answer: The exact length is $3\sqrt{3}$ metres.

infoNote

The exact answer $3\sqrt{3}$ metres is more precise than using a decimal approximation like 5.196... metres. Always express exact answers as simplified surds when possible.

Using Pythagoras to find distance between points

One of the most useful applications of Pythagoras' theorem is finding the straight-line distance between two points on a coordinate graph. This method transforms coordinate geometry problems into manageable calculations.

The process involves three key steps:

Draw a sketch showing the right-angled triangle formed by the two points
Find the lengths of the horizontal and vertical sides by subtracting coordinates
Apply Pythagoras to find the hypotenuse length, which gives you the distance

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Worked Example: Distance Between Points

Find the distance between points Q(-4, 8) and P(8, 3).

Step 1: Plot the points and draw the right-angled triangle

Step 2: Calculate the side lengths:

Horizontal side length = $8 - (-4) = 12$
Vertical side length = $8 - 3 = 5$

Step 3: Apply Pythagoras: $c^2 = 5^2 + 12^2 = 25 + 144 = 169$

Step 4: Find the distance: $c = \sqrt{169} = 13$

Answer: The distance between the two points is 13 units.

Important reminders

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Remember the Right Angle Rule: When working with Pythagoras' theorem, always ensure you're dealing with a right-angled triangle. As the saying goes, "if it's not a right angle, it's a wrong angle" - meaning the theorem simply won't work for other types of triangles.

Take time to check that your calculations are sensible. If you calculate a hypotenuse that's shorter than one of the other sides, you've made an error somewhere in your working.

For exact answers, leave your result in surd form when possible, as this maintains mathematical precision better than decimal approximations.

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Key Points to Remember:

Pythagoras' theorem only works for right-angled triangles using the formula $a^2 + b^2 = c^2$
The hypotenuse ( $c$ ) is always the longest side and sits opposite the right angle
When finding exact lengths, express answers as simplified surds rather than decimals
To find distance between coordinate points, use Pythagoras after calculating horizontal and vertical distances
Always check your answer makes sense - the hypotenuse should be longer than the other two sides

Pythagoras’ Theorem (AQA GCSE Maths): Revision Notes