Coordinates and Ratios
Coordinates, denoted by \textcolor{red}{(x, y)}, are what we use to communicate where a particular point is located on a pair of coordinate axes.
There are 3 key skills you need to know involving coordinates and ratios for GCSE Maths.
Make sure you are happy with the following topics before continuing:
Skill 1: Plotting Coordinates
Plot the point (\textcolor{red}{2},\textcolor{blue}{3})
On the xaxes we find 2, and on the y axes we find 3 and mark the point with a dot or a cross.
x = \textcolor{red}{2}
y=\textcolor{blue}{3}
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Skill 2: Finding the Midpoint of a Line
Find the midpoint of the line segment which joins points
\textcolor{Orange}{A:(2,3)} and \textcolor{blue}{B:(10,7)}
The word midpoint refers to the point which is exactly halfway between the two points in question.
Step 1: Find the half way point of the x coordinates by adding them up and dividing by 2,
\text{Midpoint of }x\text{ coordinates } = (2 + 10) \div 2 = 6
Step 2: Repeat for the y coordinates,
\text{Midpoint of }y\text{ coordinates} = (7 + 3) \div 2 = 2
Step 3: Write the values for the midpoint as a coordinate: \textcolor{red}{(6, 2)}
Skill 3: Ratios to find Coordinates
Finding a point a certain way between two points is the hardest part of this topic and involves a good knowledge of using ratios.
Example: Points A and B have coordinates (5, 16) and (3, 12) respectively.
Point C lies on the line segment between points A and B such that AC:CB = \textcolor{red}{3}:\textcolor{blue}{5}.
Find the coordinates of point C.
Step 1: Find the total parts.
\textcolor{red}{3}:\textcolor{blue}{5} which means there are \textcolor{limegreen}{8} parts in total.
The distance from A to C is \textcolor{red}{3} parts of a total \textcolor{limegreen}{8}.
Step 2: Find the x value.
Total distance in the x coordinates:
3  (5) = 8
Next we find \dfrac{\textcolor{red}{3}}{\textcolor{limegreen}{8}}of this.
\dfrac{\textcolor{red}{3}}{\textcolor{green}{8}} \times 8 = 3
Adding this to the x coordinate of A, we get
x\text{ coordinate of C } = 5 + 3 = 2
Step 3: Repeat for y
Total distance in the y coordinates:
12  16 = 4
\dfrac{\textcolor{red}{3}}{\textcolor{green}{8}} \times 4 = \dfrac{3}{2}
Adding this to the y coordinate of A, we get
y\text{ coordinate of C } = 16 + \left(\dfrac{3}{2}\right) = \dfrac{29}{2}
Therefore, the coordinates of C are \left(2, \dfrac{29}{2}\right).
You could also write this as \left(2, 14.5\right).
Take an Online Exam
Coordinates and Midpoints Online Exam
Coordinates and Ratios Online Exam
Example Questions
Question 1: Write down the coordinates of points A, B, and C seen below.
[3 marks]
A is  2 in the x direction and 2 in the y direction, so A = (2, 2).
B is  1 in the x direction and  2 in the y direction, so B = (1, 2).
C is 3 in the x direction and 0 in the y direction, so C = (3, 0).
Question 2: Find the midpoint of the line segment that joins points A and B as seen below.
[2 marks]
Point A has coordinates (2, 2).
Point B has coordinates (0, 3).
By taking the average of the x coordinates of A and B, the x coordinate of the midpoint is
\dfrac{2 + 0}{2} = 1.
By taking the average of the y coordinates of A and B, the y coordinate of the midpoint is
\dfrac{2 + 3}{2} = \dfrac{1}{2}.
Therefore, the coordinates of the midpoint are \left(1, \dfrac{1}{2}\right).
Note: it is often useful to check the graph to see if your answer looks correct.
Question 3: Points A and B have coordinates (10, 37) and (16, 1) respectively. Point C lies on the line segment between points A and B such that AC:CB = 2:7. Find the coordinates of point C.
[2 marks]
In a ratio of 2:7 there are 9 parts in total, and the distance from A to C constitutes 2 of those parts. Therefore, the distance from A to C counts for \dfrac{2}{9} of the total distance between A and B. So, we’re going to subtract the individual coordinates of A from B to find the distance in both x and y, and then we are going to add \dfrac{2}{9} of these respective distances to the coordinates of point A.
First, x coordinates: 16 (10) = 6, then
\dfrac{2}{9} \times (6) = \dfrac{12}{9} = \dfrac{4}{3}
Adding this to the x coordinate of A, we get
x\text{ coordinate of C } = 10 + \left(\dfrac{4}{3}\right) = \dfrac{34}{3}
Second, y coordinates: 1  37 = 36, then
\dfrac{2}{9} \times (36) = \dfrac{72}{9} = 8
Adding this to the y coordinate of A, we get
y\text{ coordinate of C } = 37 + (8) = 29
Therefore, the coordinates of C are \left(\dfrac{34}{3}, 29\right).
Worksheets and Exam Questions
(NEW) Coordinates & Ratios Exam Style Questions  MME
Level 35 New Official MMEDrill Questions
Mid points Ratios and Coordinates  Drill Questions
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