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Maths

Pythagoras' theorem

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Pythagoras' theorem

In a nutshell

Pythagoras' theorem is very useful when trying to find the value for the third side of a right-angled triangle. Additionally, at this level, you may have to apply the formula to finding the distance between 2 points on a graph.


Recap

Pythagoras' theorem says that:

 a2+b2=c2a^2+b^2=c^2


Where aa​ and bb​ are the two shorter sides of a right-angled triangle and cc​ is the hypotenuse.


Note: Pythagoras' theorem only works for a right-angled triangle. If there is no right-angle, then you cannot use Pythagoras' theorem.


Hypotenuse

The hypotenuse of a triangle is the longest side. For a right-angled triangle, the hypotenuse is always the side opposite the right-angle.



Distance between two points on a graph

You may be asked to find the distance between 2 given coordinates. Approaching these types of problems always requires the same method.

PROCEDURE

1.

Do a quick sketch of the graph with the given coordinates and draw a right-angled triangle with the two points.

2.

Find the lengths of the two shorter sides of the triangle.

3.

Use Pythagoras' theorem to find the length of the hypotenuse of the triangle - this is the distance.


Example

Find the distance between the points A(1,3)A (-1,3)​ and B(5,5)B(5,-5).


First, make a quick sketch:


Maths; Trigonometry; KS4 Year 10; Pythagoras' theorem


The length of the horizontal side is:

5(1)=65 - (-1) = 6


The length of the vertical side is:

3(5)=83 - (-5) = 8


The distance (labelled dd) will then satisfy Pythagoras' theorem:

62+82=d26^2 + 8^2 = d^2​​


36+64=d236+64=d^2​​


d2=100d^2 = 100​​


d=100=10\underline{d = \sqrt{100} = 10}




Want to find out more? Check out these other lessons!

Pythagoras’ theorem

Frequently Asked Questions (FAQ)

FAQs

  • Question: How do I find the distance between 2 points on a graph?

    Answer: First sketch the graph and points. Then, make a right-angled triangle with the points and use Pythagoras' Theorem.

  • Question: What is the hypotenuse of a triangle?

    Answer: The hypotenuse of a triangle is the longest side.

  • Question: When can I use Pythagoras' Theorem?

    Answer: You can only use Pythagoras' Theorem when you are given 2 sides of a right-angled triangle and want to find the third side.

Theory

Exercises

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