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Transformations

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Dec 5, 2025

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Understanding the Pythagorean Theorem

Gabriela

@gabriela_aowp

Pythagoras' theoremis one of the most useful tools in... Show more

D
Pythagaras theorem
Work out the length of BC
Give your answer to 35F.
B
10cm
A
20CM
BC = 22.4 cm.
work out length QR
Correct to 3SF
P
16cm

Finding the Hypotenuse and Shorter Sides

The Pythagoras theorem formula $a^2 + b^2 = c^2$ is your key to solving right-angled triangle problems. In this formula, $c$ represents the hypotenuse (the longest side opposite the right angle), whilst $a$ and $b$ are the two shorter sides.

When finding the hypotenuse, you simply add the squares of the two shorter sides. For example, with sides of 10cm and 20cm: $10^2 + 20^2 = 100 + 400 = 500$ . Then take the square root: $\sqrt{500} = 22.4$ cm (to 3 significant figures).

For shorter sides, you'll need to rearrange the formula to $a^2 = c^2 - b^2$ . If you know the hypotenuse is 16cm and one side is 8cm, calculate: $16^2 - 8^2 = 256 - 64 = 192$ . Taking the square root gives you $\sqrt{192} = 13.9$ cm (to 3SF).

Top Tip: Always check which side you're looking for - if it's the longest side, add the squares; if it's a shorter side, subtract them!

Complex Triangle Problems

Real-world problems often involve compound shapes where you need to apply Pythagoras' theorem multiple times. These problems might require you to find several missing lengths before reaching your final answer.

In the first example, you're working with a shape where you first find one missing side using $3.2^2 - 1.9^2 = 6.63$ , giving $\sqrt{6.63} = 2.57$ m. Then you add this to another length: $2.1 + 2.57 = 4.67$ m for the total length BC.

The second problem shows a more complex shape where you need to identify right-angled triangles within the figure. Here, you calculate $10^2 - 6^2 = 64$ , so one section is $\sqrt{64} = 8$ cm. Then for the complete length AC, you use $9^2 + 8^2 = 145$ , giving $\sqrt{145} = 12.0$ cm (to 3SF).

Remember: Break complex shapes into simple right-angled triangles, and always round your final answer to the required number of significant figures!

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Subjects

Maths

Geometry and measure

Transformations

Maths

•

605

•

Dec 5, 2025

•

2 pages

Understanding the Pythagorean Theorem

Gabriela

@gabriela_aowp

Pythagoras' theorem is one of the most useful tools in maths for finding missing lengths in right-angled triangles. Whether you're calculating the hypotenuse or finding a shorter side, mastering this theorem will help you tackle geometry problems with confidence.

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Finding the Hypotenuse and Shorter Sides

Top Tip: Always check which side you're looking for - if it's the longest side, add the squares; if it's a shorter side, subtract them!

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Complex Triangle Problems

Remember: Break complex shapes into simple right-angled triangles, and always round your final answer to the required number of significant figures!