Coordinate geometry · Ages 14–15
The distance formula
The run and the rise are the legs, AB the hypotenuse.
How to do it
Method
The run and the rise are the legs, AB the hypotenuse
A(1, 2), B(4, 6): 32 + 42 = 5
Another way
Draw the right triangle
3 squares across, 4 up: the hypotenuse is 5
In full
The distance between A(x₁, y₁) and B(x₂, y₂) is AB = (x₂ − x₁)2 + (y₂ − y₁)2. It is Pythagoras' theorem: go from A across to straight below (or above) B, then up or down to B; the distance across and the distance up are the legs of a right-angled triangle, and AB is its hypotenuse. A(1, 2), B(4, 6): across 4 − 1 = 3, up 6 − 2 = 4, AB = 32 + 42 = 25 = 5. With negative coordinates, bracket them as you subtract: A(−2, 3), B(4, −5): 4 − (−2) = 6, −5 − 3 = −8, (−8)2 = 64, AB = 36 + 64 = 10. Either point can come first: the squares are the same. When the root isn't whole, leave it as □ or take an approximation. Or draw the right triangle on squared paper, count its legs and use Pythagoras: the same answer. Two points on one horizontal line are as far apart as their x-coordinates differ; on one vertical line, as their y-coordinates differ.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
In the picture, from A to B is units across and units up or down, so AB = .
Fill in each blank
Show the answer and working
Answer: 8, 6, 10
8 across and 6 up or down are the legs of a right triangle and AB its hypotenuse: 82 + 62 = 10.
Example 2
The distance between A(−2, −4) and B(3, 8) is 14.
True or false?
Show the answer and working
Answer: False
The run is 5 and the rise 12: AB = 52 + 122 = 13.
Example 3
A(−1, −8), B(−9, k), with B above A and AB = 10. k =
Show the answer and working
Answer: −2
The run is 8, so the rise = 102 − 82 = 6; B is above, so k = −8 + 6 = −2.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.The distance between A(−3, 0) and B(9, −5) is 15.
True or false?
Answer:
2.In the picture, from A to B is units across and units up or down, so AB = .
Fill in each blank
Answer:
3.How far apart are A(−4, −1) and B(3, −6)?
- A. 74
- B. 74
- C. 12
- D. 50
Choose the right answer
Answer:
4.A(3, 3), B(4, −2). AB =
Answer:
5.A(−5, −10), B(−10, k), with B above A and AB = 13. k =
Answer:
All the answers
- 1. False
- 2. 4, 3, 5
- 3. A. 74
- 4. 26
- 5. 2
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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