Coordinates and Ratios Worksheets | Questions and Revision | MME

# Coordinates and Ratios Worksheets, Questions and Revision

Level 6 Level 7

## 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 $x$axes 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

$A$ 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 $C = (3, 0)$.

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 $\left(-1, -\dfrac{1}{2}\right)$.

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 $\left(-\frac{34}{3}, 29\right)$.

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