Dividing a line in a given ratio Revision Notes for AQA GCSE Further Maths

Dividing a line in a given ratio

Introduction to the concept

When working with coordinate geometry, you'll often need to find the coordinates of a point that divides a line segment in a specific ratio. This is a powerful technique that combines ratio work with coordinate geometry to solve various problems.

The key idea is that if you know the coordinates of two endpoints of a line segment, you can calculate where any point lies along that line if you're told the ratio in which it divides the segment.

Understanding through coordinates

Let's start by looking at a practical example to understand how this works. Consider two points $A(4, 7)$ and $B(19, 27)$ on a coordinate plane.

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When point C divides the line AB in a specific ratio, we can use the horizontal and vertical distances to help us find C's coordinates. In the diagram above, we can see how the horizontal distance $AE = 19 - 4 = 15$ and the vertical distance $BE = 27 - 7 = 20$ .

If C divides line AB in the ratio $2:3$ , this means $AC:CB = 2:3$ . This ratio relationship applies to both the horizontal and vertical components of the line segment.

The section formula

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The Section Formula

The section formula is the key tool for solving these problems. When a point C divides a line segment AB internally in the ratio $p:q$ , where A has coordinates $(x_1, y_1)$ and B has coordinates $(x_2, y_2)$ , the coordinates of C can be found using:

$C = \left(\frac{qx_1 + px_2}{p + q}, \frac{qy_1 + py_2}{p + q}\right)$

This formula works because the coordinates of the dividing point are essentially weighted averages of the endpoint coordinates, where the weights are determined by the ratio.

Step-by-step method

Here's a systematic approach to solve these problems:

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Method for Finding Dividing Point Coordinates:

Identify the given information: Note the coordinates of the two known points and the ratio in which the line is divided
Determine which ratio applies: Check whether you're given $AC:CB$ or $AC:AB$ (these are different!)
Apply the section formula: Substitute the coordinates and ratio values into the formula
Calculate separately: Find the x-coordinate and y-coordinate of the dividing point
Verify your answer: Check that your result makes sense geometrically

Worked example 1: Finding coordinates with internal division

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Worked Example: Internal Division

Problem: AB is a straight line where A is $(-3, 2)$ and B is $(4, -19)$ . Point C lies on AB such that $AC:CB = 4:3$ . Calculate the coordinates of C.

Solution:

Here we have $A(-3, 2)$ and $B(4, -19)$ , with the ratio $p:q = 4:3$ .

Using the section formula:

For the x-coordinate: $x = \frac{qx_1 + px_2}{p + q} = \frac{3(-3) + 4(4)}{4 + 3} = \frac{-9 + 16}{7} = \frac{7}{7} = 1$

For the y-coordinate: $y = \frac{qy_1 + py_2}{p + q} = \frac{3(2) + 4(-19)}{4 + 3} = \frac{6 - 76}{7} = \frac{-70}{7} = -10$

Therefore, C is at (1, -10).

Worked example 2: Finding coordinates when given partial information

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Worked Example: Finding Unknown Endpoint

Problem: In the figure above, PQR is a straight line with $P(2, -3)$ and $Q(-2, 5)$ . The ratio $PQ:QR = 2:5$ . Find the coordinates of R.

Solution:

This problem is slightly different because we need to find R, and we're given the ratio $PQ:QR$ rather than $PQ:PR$ .

Since $PQ:QR = 2:5$ , we can work with $Q(-2, 5)$ and the unknown R, knowing that $P(2, -3)$ relates to them in this specific ratio.

Let R have coordinates $(x_2, y_2)$ . We know that Q divides PR in the ratio $2:5$ .

Working with the section formula where Q divides PR in ratio $2:5$ :

For the x-coordinate: $-2 = \frac{5(2) + 2x_2}{2 + 5}$ $-2 = \frac{10 + 2x_2}{7}$ $-14 = 10 + 2x_2$ $-24 = 2x_2$ $x_2 = -12$

For the y-coordinate: $5 = \frac{5(-3) + 2y_2}{2 + 5}$ $5 = \frac{-15 + 2y_2}{7}$ $35 = -15 + 2y_2$ $50 = 2y_2$ $y_2 = 25$

Therefore, R is at (-12, 25).

Key formulas and definitions

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Essential Definitions and Formulas:

Internal Division: When a point lies between two given points on a line segment.

Section Formula: $C = \left(\frac{qx_1 + px_2}{p + q}, \frac{qy_1 + py_2}{p + q}\right)$ where C divides AB in ratio $p:q$ .

Ratio Notation: $AC:CB = p:q$ means the distance from A to C compared to the distance from C to B is in the ratio $p$ to $q$ .

Common exam tips and pitfalls

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Watch out for these common mistakes:

Ratio confusion: Make sure you understand which segments the ratio refers to. $AC:CB$ is different from $AC:AB$ .
Formula mix-up: In the section formula, the coordinates get "cross-multiplied" with the ratios. The coordinate of the first point multiplies with the second ratio number.
Sign errors: Be extra careful with negative coordinates - it's easy to make arithmetic mistakes.
Checking your work: Your answer should make geometric sense. If C divides AB in ratio $2:1$ , then C should be closer to B than to A.

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Key Points to Remember:

The section formula gives you the coordinates of any point that divides a line segment in a given ratio
When C divides AB in ratio $p:q$ , use: $C = \left(\frac{qx_1 + px_2}{p + q}, \frac{qy_1 + py_2}{p + q}\right)$
The coordinates are weighted averages - the ratios act as weights for the endpoint coordinates
Always double-check which segments your ratio refers to ( $AC:CB$ vs $AC:AB$ makes a big difference)
Practice with both positive and negative coordinates to avoid arithmetic errors

Dividing a line in a given ratio (AQA GCSE Further Maths): Revision Notes