The distance between two points (AQA GCSE Further Maths): Revision Notes

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Finding the distance between two points

When working with coordinate geometry, you'll often need to find the straight-line distance between two points on a coordinate plane. This is where your knowledge of Pythagoras' theorem becomes incredibly useful.

Understanding the concept

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The key insight is that the distance between any two points can be found by creating a right-angled triangle. The distance you want to find becomes the hypotenuse of this triangle, while the horizontal and vertical distances between the points form the two shorter sides.

This approach transforms a coordinate geometry problem into a familiar Pythagoras theorem application, making it much more manageable to solve.

Worked example

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Worked Example: Finding Distance Between Two Points

Find the distance between points P(3, 1) and Q(6, 5).

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Step 1: Find the horizontal distance The horizontal distance is the difference between the x-coordinates: Horizontal distance = 6−3=36 - 3 = 3

Step 2: Find the vertical distance
The vertical distance is the difference between the y-coordinates: Vertical distance = 5−1=45 - 1 = 4

Step 3: Apply Pythagoras' theorem Now we have a right-angled triangle where:

  • One side = 3 (horizontal distance)
  • Other side = 4 (vertical distance)
  • Hypotenuse = distance PQ (what we want to find)

Using Pythagoras' theorem: a2+b2=c2a^2 + b^2 = c^2

PQ=32+42=9+16=25=5PQ = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

So the distance between P and Q is 5 units.

The distance formula

We can generalise this method into a formula that works for any two points.

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If you have two points:

  • Point P with coordinates (x_1,y_1)(x\_1, y\_1)
  • Point Q with coordinates (x_2,y_2)(x\_2, y\_2)

Then the distance between them is:

Distance=(x_2−x_1)2+(y_2−y_1)2\text{Distance} = \sqrt{(x\_2 - x\_1)^2 + (y\_2 - y\_1)^2}

This formula essentially does the same three steps we used in the worked example, but in one go.

Key steps to remember

When finding the distance between two points, follow these essential steps:

  1. Identify your coordinates: Make sure you know which point is which and label them clearly
  2. Calculate horizontal distance: Subtract the x-coordinates (x_2−x_1)(x\_2 - x\_1)
  3. Calculate vertical distance: Subtract the y-coordinates (y_2−y_1)(y\_2 - y\_1)
  4. Square both distances: This ensures positive values
  5. Add the squares together
  6. Take the square root: This gives you the final distance

Common exam tips

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Critical Exam Strategies:

  • Always draw a diagram if one isn't provided - it helps you visualise the problem
  • Be careful with the order of subtraction - it doesn't matter which way round you subtract as you're squaring the result anyway
  • Check your answer makes sense by estimating - the distance should be longer than either the horizontal or vertical distance alone
  • Remember that distance is always positive
  • Practice using the formula directly once you're comfortable with the step-by-step method

Summary

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Key Points to Remember:

  • The distance between two points uses Pythagoras' theorem by creating a right-angled triangle
  • The horizontal distance is the difference between x-coordinates
  • The vertical distance is the difference between y-coordinates
  • The distance formula is: (x_2−x_1)2+(y_2−y_1)2\sqrt{(x\_2 - x\_1)^2 + (y\_2 - y\_1)^2}
  • Always check your answer makes geometric sense

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