# Coordinates and Ratios

## Coordinates and Ratios Revision

**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 x-axes 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}

**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)}

You can also find the midpoint of a line segment, if the two points at either end are drawn on a grid. , just by looking at it.

**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).

## Coordinates and Ratios 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).