1. GCSE/KS3 :: Coordinates
Fundamentals
GCSE/KS3 :: Coordinates
@DrFrostMaths
Objectives: Understand coordinates in all four ‘quadrants’ and solve problems involving coordinates.
Last modified: 10th January 2020

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3. STARTER :: Dimensions
For each of these vehicles, think about the directions in which each vehicle can move, and what restrictions there might be.
At any given moment in time, a train/tram can only move in one direction (or backwards in that direction).
We say that a train can move in 1 dimension.
At any given moment, a car can move forwards/backwards, and also left/right (or a combination of the two), but not up/down*.
We say that a car can move in 2 dimensions.
A plane however can move forwards/backwards, left/right, and up/down!
We say that a plane can move in 3 dimensions.
* Although a car can go uphill, it can’t usually move up/down on the spot!
Fro Note: While the movement of trains and cars are limited to “1D” and “2D”, they both occupy 3D space.

4. Number Lines
You may be familiar with a number line. It can be used to indicate the position of something in one direction (i.e. “in 1 dimension”) km ?
You’ve ended up going 3km AWAY from school!
You’re a distance of 0km from home, i.e. haven’t left! ? ?
You’re 4km from home (in the direction of your school)
Suppose this number line indicated your position (in kilometres) on your journey to school. What would each arrow mean?

5. Extending to up/down 𝑦
We can add a second direction, allowing us to now describe where something is, both how far left/right it is, but also how much up/down it is.
This vertical ‘number line’ is known as the 𝑦-axis. 5 4 3
This horizontal ‘number-line’ is known as the 𝑥-axis. 2 1 𝑥 𝑂 -1
The point where the two axes meet (i.e. where we haven’t moved left/right nor up/down) is known as the origin.
We put the letter 𝑂 there. -2 -3
We’ll just look at positive values for the moment! -4

6. (6,4) Coordinates in the first quadrant
How can we describe where the point is?
We’d say the coordinates (i.e. position) of this point are:
(6,4) 𝑦 5 4
𝑥 value
𝑦 value 3
Don’t forget the brackets!
We first work out how far left/right we need to go, starting from the origin.
i.e. “How far along the 𝑥-axis”. 2
We then work out how far up/down we need to go, i.e. “How far up the 𝑦-axis” 1 𝑥 𝑂 1 2 3 4 5 6 7 8
Fro Tip: To remember whether we move left/right or up/down first, notice that 𝑥 and 𝑦 are in alphabetical order. So we go along the 𝑥 number-line first, then the 𝑦.

7. More Examples ? 𝐴: (𝟑,𝟐) ? 𝐵: (𝟓,𝟓) 𝐶: (𝟏,𝟒) ? ? 𝐷: (𝟕,𝟎) 𝐸: (𝟎,𝟑) ?
What are the coordinates of each point? 𝑦 ?
𝐴: (𝟑,𝟐) 𝐵 5 ?
𝐵: (𝟓,𝟓) 𝐶 4
𝐶: (𝟏,𝟒) ? 𝐸 3 ?
𝐷: (𝟕,𝟎) 𝐴 2
𝐸: (𝟎,𝟑) ? 1
For D, we haven’t moved up/down, so the 𝑦 value is 0. 𝐷 𝑥 𝑂 1 2 3 4 5 6 7 8

8. Test Your Understanding𝑦
What are the coordinates of points 𝐴, 𝐵 and 𝐶?
𝑨 𝟕,𝟐 𝑩 𝟔,𝟓 𝑪 𝟎,𝟑
Add the point (1,0) to the diagram and label it 𝐷.
Join 𝐴 to 𝐵 to 𝐶 to 𝐷 to 𝐴 with straight lines. What shape is 𝐴𝐵𝐶𝐷?
A rectangle. 1 𝐵 5 ? 4 𝐶 3 2 𝐴 2
? Reveal 1 3 𝐷 𝑥 𝑂 1 2 3 4 5 6 7 8 ?

9. Exercise 1 ? 𝐴: (𝟒,𝟑) 𝐵: (𝟕,𝟒) ? 𝐶: (𝟐,𝟓) ? 𝐷: (𝟔,𝟎) ? 𝐸: (𝟎,𝟒) ?
Determine the coordinates of each point. 1 𝑦 𝐶 5
𝐴: (𝟒,𝟑) ? 𝐸 𝐵
𝐵: (𝟕,𝟒) ? 4 𝐴
𝐶: (𝟐,𝟓) ? 3
𝐷: (𝟔,𝟎) ? 2
𝐸: (𝟎,𝟒) ? 1 𝐷 𝑥 𝑂 1 2 3 4 5 6 7 8

10. Exercise 1 𝐴: (𝟐,𝟏) 𝐵: (𝟑,𝟓) 𝐶: (𝟎,𝟏) 𝐷: (𝟖,𝟎) 𝐸: (𝟒,𝟒) 2
Mark the following points on the graph. 𝑦 𝐵 5
? Reveal
𝐴: (𝟐,𝟏) 𝐸
? Reveal 4
𝐵: (𝟑,𝟓)
𝐶: (𝟎,𝟏)
? Reveal 3
𝐷: (𝟖,𝟎)
? Reveal 2 𝐶 𝐴
𝐸: (𝟒,𝟒)
? Reveal 1 𝐷 𝑥 𝑂 1 2 3 4 5 6 7 8

11. Exercise 1
Decipher the secret message! 3 𝑦 𝐴 𝐵 𝐶 𝑃 𝑄 5 𝐷 𝐸 𝐹 𝑅 𝑆 4 𝐺 𝐻 𝐼 𝑇 𝑈 3 𝐽 𝐾 𝑉 𝑊 2 𝑋 𝑌 𝐿 𝑀 𝑁 1 𝑂 𝑍 𝑥 𝑂 1 2 3 4 5 6 7 8
5,3 , 5,1 , 7,3 , 4,3 , 2,4 , 4,0 , 7,2 , 2,4 ,(5,1) ?
IN THE OVEN

12. Coordinates in all four quadrants𝑦
2nd Quadrant 4
1st Quadrant 3 2
We’ve so far used coordinates where the 𝑥 and 𝑦 value are both positive.
This region is known as the 1st quadrant. 1 𝑥
3rd Quadrant -3 -2 -1 𝑂
4th Quadrant 1 2 3 4 -1 -2
Fro Memory Tip: The quadrants are numbered anticlockwise. -3

13. Examples 𝑦 4
What are the coordinates of 𝐴 and 𝐵? 𝐴 3 2
𝐴( , ) −2 3 1
As before we first see how far left/right on the 𝑥-axis we move. This time it’s a negative number. 𝑥 -3 -2 -1 𝑂 1 2 3 4 𝐵 -1
𝐵( , ) −3 −1 -2 -3
You can see we’re level with the -1 on the 𝑦-axis (the 𝑦 ‘number line’)

14. Test Your Understanding𝑦 𝐺 𝐴 1
Identify the coordinates of the following points. 4 3
𝑨 −𝟏,𝟒 𝑩 −𝟐,𝟎 𝑪 𝟐,−𝟏 𝑫 −𝟐,−𝟑 𝑬 𝟒,−𝟐 𝑭(𝟎,−𝟑) ? ? 2 ? ? 1 ? 𝐵 𝑥 ? -3 -2 -1 𝑂 1 2 3 4 𝐶
Add points to your diagram at the following positions:
𝑮 −𝟑,𝟒 𝑯 𝟒,−𝟑 -1 2 𝐸 -2
? Reveal 𝐷 𝐹 𝐻
? Reveal -3

15. Exercise 2 𝑦 1
Identify the coordinates of the following points. 4 3
𝑨 𝟐,𝟏 𝑩 −𝟏,𝟐 𝑪 −𝟐,−𝟏 𝑫 𝟏,−𝟐 𝑬 𝟐,𝟎 𝑭 𝟎,𝟐 𝑮 −𝟐,𝟎 𝑯(𝟎,−𝟐) ? ? 𝐵 𝐹 2 ? 𝐴 ? 1 ? 𝐺 𝐸 𝑥 ? -3 -2 -1 𝑂 1 2 3 4 ? 𝐶 -1 ? 𝐻 𝐷 -2 -3

16. Exercise 2 𝑦 2 a
Draw the following points on the provided axes. 4 𝐶 3
? Reveal
𝑨 𝟏,𝟐 𝑩 −𝟑,𝟐 𝑪 −𝟏,𝟑 𝑫 −𝟏,−𝟐
? Reveal 𝐵 𝐴 2
? Reveal
? Reveal 1 𝑥
If you join the points together, what shape does this make? A kite. b -3 -2 -1 𝑂 1 2 3 4 -1 ? 𝐷 -2 -3

17. Exercise 2 𝑦 3 4
Break the code by working out the letter for each coordinate! 𝐴 𝐵 𝐶 𝑃 3 𝐷 𝐸 𝐹 𝑅 2 𝐺 𝐻 𝐼 𝑇 1 𝐽 𝐾 𝑉 𝑥 -3 -2 -1 𝑂 1 2 3 4 𝑋 𝐿 𝑀 𝑁 -1 𝑂 -2 𝑄 𝑆 𝑈 𝑊 𝑌 𝑍 -3
−𝟏,𝟐 , 𝟎,−𝟐 , −𝟐,−𝟏 , −𝟐,−𝟏 , 𝟎,−𝟐 , 𝟏,−𝟑 , 𝟑,𝟏 , 𝟎,𝟏 , −𝟐,𝟐 , −𝟐,−𝟏 , −𝟐,−𝟏 , −𝟑,𝟑 , −𝟏,−𝟏 ,(−𝟑,𝟑)
FOLLOW THE LLAMA ?

18. Problem Solving
Sometimes you might need to work what points is halfway between two other points (known as the midpoint), or where you would put a point to make a certain shape. 𝑦 4 𝐴 𝑫 3
The point 𝐷 is placed so that the shape 𝐴𝐵𝐶𝐷 is a rectangle.
What are the coordinates of 𝐷? 2 1 -3 -2 -1 𝑂 1 2 3 4 𝑥
We can see by observation that putting 𝐷 here would complete the rectangle. 𝐵 -1 𝐶 -2
𝑫(𝟐,𝟑) -3

19. Another Example
Determine the coordinates of the midpoint of 𝐴 and 𝐵. 𝑦 4
The midpoint means the point halfway between 𝑨 and 𝑩.
Again, we can see just by observation where that point is. 𝐴 3 𝑴 2 𝐵 1 𝑥
(𝟎,𝟐) -3 -2 -1 𝑂 1 2 3 4 𝐵 -1 -2 -3

20. Test Your Understanding𝑦 a
𝐴 −2,3 , 𝐵 2,3 ,𝐶(3,1) are three of the points of a parallelogram 𝐴𝐵𝐶𝐷. Determine the coordinates of 𝐷.
(−𝟏,𝟏)
Determine the coordinates of the midpoint of 𝐶𝐷.
(𝟏,𝟏) 4 𝐴 𝐵 3 ? 2 𝑫 𝐶 1 b 𝑥 -3 -2 -1 𝑂 1 2 3 4 ?
Fro Tip: Because opposite sides of a parallelogram are the same, we can copy the movement at one end (i.e. 1 right and 2 down) at the other end (from 𝐴) -1 -2 -3

21. Exercise 3 ? 1 [Edexcel GCSE June2007-1F Q14cii]
On the grid, the point 𝐷 is such that 𝐴𝐵𝐶𝐷 is a rectangle.
Write down the coordinates of 𝐷.
(−𝟒,−𝟐) ?

22. Exercise 3 ? 2 [Edexcel GCSE Nov2012-1F Q11c]
P, Q and R are three vertices of a parallelogram.
Write down the coordinates of the fourth vertex of this parallelogram.
(𝟑,−𝟐) ?

23. Exercise 3 ? [Edexcel IGCSE(9-1) June 2018 2F Q4c] 3
The diagram shows points A, B and C on a square grid. On the grid, mark with a cross (X) the point 𝐷 so that 𝐴𝐵𝐶𝐷 is a parallelogram, and identify its coordinates.
(−𝟏,−𝟏) 3 ? 𝐷

24. Exercise 3 ? 4 [Edexcel GCSE Nov2016-1F Q7]
The points (–3, –1), (–2, 2) and (3, 2) are three vertices of a parallelogram.
Find the coordinates of the fourth vertex of the parallelogram.
(𝟐,−𝟏) ?

25. Exercise 3 ? 5 [KS2 SATs 2016 Sample Paper 3 Q20a]
Here are two identical shaded triangles on coordinate axes.
Write the coordinates of the point A.
(𝟏𝟐,𝟎) ?

26. Exercise 3 ? 6 𝑁 𝑀 [Edexcel GCSE Nov2007-2F Q11d]
N is the point (–3, 2). M is another point.
The 𝑥 coordinate of M is the same as the 𝑥 coordinate of N. The 𝑦 coordinate of M is the same as the 𝑦 coordinate of B.
Write down the coordinates of the point M.
(−𝟑,𝟎) ?

27. Exercise 3 ? 7 𝐵 [OCR GCSE Nov 2015 1F Q3c Edited]
Points 𝐴 and 𝐶 are plotted on a coordinate grid.
The point B had coordinates −4,−3 .
What is the mathematical name of triangle 𝐴𝐵𝐶?
Scalene ?

28. Exercise 3 ? 8 𝐵 [OCR GCSE June 2016 2F Q2c Edited]
Points A and B are marked on this grid.
Point C has coordinates 5,−2 .
What type of triangle is ABC?
Isosceles ?

29. Exercise 3 ? 9 [OCR GCSE Nov 2016 1F Q3bii Edited]
Points 𝑃, 𝑄 and 𝑅 are shown on this one-centimetre square grid.
𝑃𝑄 is the diameter of a circle. The circle has its centre at point 4,3 . Find the radius of this circle.
Radius is 3 ?

30. Exercise 3 10 ?
[Edexcel GCSE(9-1) June F Q24, June H Q6] A pattern is made from four identical squares. The sides of the squares are parallel to the axes. Point 𝐴 has coordinates 6,7 Point 𝐵 has coordinates 38,36 Point 𝐶 is marked on the diagram.
Work out the coordinates of 𝐶.

31. Exercise 3 11
[JMC 2007 Q13] Points 𝑃 and 𝑄 have coordinates 1,4 and 1,−2 respectively. For which of the following possible coordinates of point R would triangle 𝑃𝑄𝑅 not be isosceles?
[   ]  −5,4 [   ] 7,1 [   ]  −6,1 [   ]  −6,−2 [   ]  7,−2 ?
(−𝟔,−𝟐)

32. Exercise 3 ? 12 [OCR GCSE(9-1) June 2018 2F Q16a]
Four identical trapeziums are placed on a coordinate grid as shown.
Write down algebraic expressions for the coordinates of point 𝑃.
(𝒂, 𝒂−𝒃) ?