Coordinates and Ratio Revision Notes for AQA GCSE Maths

Coordinates and ratio

When working with graphs and coordinate geometry, understanding how to find points along line segments is essential. A line segment is simply the portion of a line that exists between two specific endpoints, rather than continuing infinitely in both directions.

Finding the midpoint using the average of endpoints

The midpoint of a line segment sits exactly halfway between the two endpoints. To calculate the coordinates of this midpoint, you need to work with each coordinate separately and find their average.

Method for finding midpoints

To determine the midpoint coordinates, follow these steps:

For the x-coordinate: Take the x-coordinates of both endpoints, add them together, then divide the result by 2
For the y-coordinate: Take the y-coordinates of both endpoints, add them together, then divide the result by 2

This averaging process gives you the exact centre point between the two endpoints.

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The midpoint formula can be written mathematically as: If we have points $A(x_1, y_1)$ and $B(x_2, y_2)$ , then the midpoint $M$ has coordinates:

$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$

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Worked Example: Finding the Midpoint

Let's say we have points A at coordinates $(7, 4)$ and B at coordinates $(-1, -2)$ . To find the midpoint M:

Step 1: Calculate the x-coordinate of the midpoint

Add the x-coordinates: $7 + (-1) = 6$
Divide by 2: $6 ÷ 2 = 3$

Step 2: Calculate the y-coordinate of the midpoint

Add the y-coordinates: $4 + (-2) = 2$
Divide by 2: $2 ÷ 2 = 1$

Therefore, the midpoint M has coordinates $(3, 1)$ .

Using ratios to find coordinates

Ratios provide a powerful way to locate points that lie anywhere along a line segment, not just at the midpoint. When you know that a point divides a line segment in a particular ratio, you can calculate its exact coordinates.

Understanding ratios on line segments

A ratio tells you how far along a line segment a point is positioned. For example, if point C divides line segment AB in the ratio $4:3$ , this means:

The distance from A to C is 4 parts
The distance from C to B is 3 parts
The total distance from A to B is $4 + 3 = 7$ parts

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When working with ratios, always remember that the ratio describes the relationship between the segments on either side of the point. The first number in the ratio corresponds to the distance from the starting point to the dividing point.

Method for using ratios

To find coordinates using ratios, follow these steps:

Calculate the differences: Find the difference between the x-coordinates and y-coordinates of the endpoints
Apply the ratio: Use the ratio to determine what fraction of each difference to add to the starting point
Add to the starting coordinates: Add these calculated amounts to the coordinates of the starting point

Practical applications

These coordinate and ratio techniques are particularly useful when:

Finding centres of geometric shapes
Locating points at specific distances along paths
Solving problems involving proportional relationships
Working with vectors and transformations

The key is to remember that ratios describe proportional relationships, whilst the midpoint represents a special case where the ratio is always $1:1$ .

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Key Points to Remember:

The midpoint is found by averaging the coordinates of the endpoints
For midpoints: add the coordinates and divide by 2
Ratios tell you the proportional position of a point along a line segment
When using ratios, calculate the coordinate differences first, then apply the fraction
Always add your calculated values to the starting point's coordinates
The midpoint is just a special case of ratio division where the ratio is $1:1$

Coordinates and Ratio (AQA GCSE Maths): Revision Notes