What you need to know
Coordinates – Basics
Coordinates, denoted by (x, y), are what we use to communicate where a particular point is located on a pair of coordinate axes.
Plot the point (2,3
we know:
(2,3) = (x,y)
so
x = 2
y=3
On the xaxes we find 2 and on the y axes we find 3 and mark our point.
Coordinates – Finding the Midpoints
Find the midpoint of the line segment which joins points A:(2, 3) and B:(10, -7).
The word midpoint refers to the point which is exactly halfway between the two points in question.
Step 1: There are 2 x coordinates, 2 and 10.
Determine the half way point is to add them up and divide by 2 (in other words, take the mean of the two points).
\text{Midpoint of }x\text{ coordinates } = (2 + 10) \div 2 = 6
Step 2: Repeat for the 2 y coordinates are 3 and -7. We’ll do exactly the same thing for these.
\text{Midpoint of }y\text{ coordinates} = (-7 + 3) \div 2 = -2
Therefore, the coordinates of the midpoint are (6, -2).
Ratios to find coordinates
Points A and B have coordinates (3, 12) and (-5, 16) respectively. Point C lies on the line segment between points A and B such that AC:CB = 5:3.
Find the coordinates of point C.
The ratio is 5:3 which means there are 8 parts in total.
The distance from A to C is 5 parts or a total 8.
First, lets find the total distance in the latex]x[/latex] coordinates:
-5 - 3 = -8
Next we find \dfrac{5}{8}^{\text{th}} of the this.
\frac{5}{8} \times -8 = -5
Adding this to the x coordinate of A, we get
x\text{ coordinate of C } = 3 + (-5) = -2
Second, we repeat for y
12 - 16 = -4
\frac{5}{8} \times -4 = -\frac{5}{2}
Adding this to the y coordinate of A, we get
y\text{ coordinate of C } = 16 + \left(-\dfrac{5}{2}\right) = \dfrac{27}{2}
Therefore, the coordinates of C are \left(-2, \frac{27}{2}\right).
You could also write this as \left(-2, 13.5\right)
Example Questions
1) Write down the coordinates of points A, B, and C seen below.
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).
2) Find the midpoint of the line segment that joins points A and B as seen below.
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
\frac{-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).
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.
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(-\frac{34}{3}, 29\right).
Worksheets and Exam Questions
(NEW) Coordinates & Ratios Exam Style Questions - MME
Level 3-5Mid points Ratios and Coordinates - Drill Questions
Level 1-3Videos
Coordinates and Ratios Q1
GCSE MATHSCoordinates and Ratios Q2
GCSE MATHSCoordinates and Ratios Q3
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