Coordinate geometry in the (x, y) plane

Q: What is the formula for the distance $d$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$?

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Q: How do you find the midpoint $M$ of a line segment connecting two points?

Average the coordinates: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.

Q: What is the formula for the gradient $m$ of a line passing through two points?

$m = \frac{y_2 - y_1}{x_2 - x_1}$ (also known as "Rise over Run").

Q: What are the gradients of horizontal and vertical lines?

A horizontal line has a gradient of $0$, and a vertical line has an undefined gradient.

Q: What condition must be met for two lines to be parallel?

Two lines are parallel if they have the same gradient: $m_1 = m_2$.

Q: What is the relationship between the gradients of two perpendicular lines?

Their product is $-1$ ($m_1 \times m_2 = -1$), meaning one is the negative reciprocal of the other.

Q: What is the gradient-intercept form of a straight line equation?

$y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept.

Q: What is the point-gradient formula for finding the equation of a line?

$y - y_1 = m(x - x_1)$, where $m$ is the gradient and $(x_1, y_1)$ is a point on the line.

Q: What is the general form of a line equation and what are its standard constraints?

$ax + by + c = 0$, where $a$, $b$, and $c$ must be integers.

Q: How do you find the coordinates of the point where two lines intersect?

Solve the equations of the two lines as simultaneous equations .

Practice

Pure Mathematics (YPM01) · Pearson Edexcel International A Level · 6 min read · Updated: Sep 6, 2026

Coordinate Geometry in the (x, y) Plane: Comprehensive Study Notes

Welcome to the essential world of Coordinate Geometry! This chapter is where we give algebra a visual home, allowing us to describe geometric shapes and relationships using numbers. Think of it as the mathematical equivalent of using GPS coordinates to navigate.
Mastering these tools—distance, gradient, and equations of lines—is absolutely fundamental. They form the backbone for many later topics in Pure Mathematics.

Don't worry if the formulas look complicated! We will break them down, and you’ll find that they are actually based on simple concepts you already know, like Pythagoras’ theorem.

1. Basic Tools: Distance and Midpoint

Before we tackle lines, we need reliable methods for measuring space between two points, $A(x_1, y_1)$ and $B(x_2, y_2)$.

1.1 The Distance Between Two Points

Finding the distance is simply applying the Pythagorean Theorem. If you draw a right-angled triangle between the two points, the distance is the hypotenuse.

Key Formula:

Distance $d = \sqrt{(\text{change in } x)^2 + (\text{change in } y)^2}$

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Tip for struggling students: It doesn't matter which point you label $(x_1, y_1)$ and which you label $(x_2, y_2)$. Because we square the differences, a negative sign will disappear anyway! Just be consistent (e.g., always subtract the coordinates of the first point from the coordinates of the second point).

1.2 The Midpoint of a Line Segment

The midpoint is the exact middle point of the line segment. To find the middle, we simply average the $x$-coordinates and average the $y$-coordinates.

Analogy: If two friends live at different addresses, the midpoint is the coffee shop exactly halfway between them.

Key Formula:

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

Common Mistake to Avoid: Accidentally calculating the difference in the coordinates instead of the sum. Remember, midpoint uses addition.

Key Takeaway: The distance formula is Pythagoras in disguise, and the midpoint formula is just finding the average position.

2. The Gradient (Slope) of a Line

The gradient, often denoted by $m$, measures the steepness and direction of a line.

2.1 Calculating the Gradient

The gradient is calculated as the change in the vertical distance ($y$) divided by the change in the horizontal distance ($x$).

Mnemonic: Gradient is "Rise over Run."

Key Formula:

$$m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$

2.2 Interpreting the Gradient

Positive Gradient ($m > 0$): The line slopes up from left to right (like climbing a hill).
Negative Gradient ($m < 0$): The line slopes down from left to right (like sliding down a hill).
Zero Gradient ($m = 0$): This is a horizontal line ($y = c$).
Undefined Gradient: This is a vertical line ($x = c$). You cannot divide by zero if $x_2 - x_1 = 0$.

3. Relationships Between Lines (Parallel and Perpendicular)

These relationships are critical for geometric proofs and calculations. They depend entirely on the gradients.

3.1 Parallel Lines

Two lines are parallel if they run alongside each other and never intersect. This happens only if they have exactly the same steepness.

Rule: If Line 1 has gradient $m_1$ and Line 2 has gradient $m_2$, then they are parallel if:

$$m_1 = m_2$$

3.2 Perpendicular Lines

Two lines are perpendicular if they intersect at a 90-degree angle (a right angle).

Rule: If Line 1 has gradient $m_1$ and Line 2 has gradient $m_2$, then they are perpendicular if:

$$m_1 \times m_2 = -1$$

Or, more practically, $m_2$ is the negative reciprocal of $m_1$.

Step-by-Step for Negative Reciprocal:

Reciprocate: Flip the fraction. If $m_1 = 3/4$, the reciprocal is $4/3$.
Negate: Change the sign. If $m_1$ was positive, $m_2$ must be negative, and vice versa.

Example: If the first gradient $m_1 = 5$, the perpendicular gradient $m_2 = -1/5$.

Quick Review Box: Perpendicular Trick
To find the perpendicular gradient, always "Flip it and Swap the sign!"

4. Finding the Equation of a Straight Line

An equation is a rule that every point $(x, y)$ on the line must satisfy. You generally need two things to find a line's equation: a point and the gradient.

4.1 The Gradient-Intercept Form

This is often the most familiar form.

$$y = mx + c$$

$m$ is the gradient.
$c$ is the y-intercept (where the line crosses the y-axis, at the point $(0, c)$).

4.2 The Point-Gradient Form (The Most Useful Tool)

This form is the fastest way to write the equation if you know the gradient $m$ and one point $(x_1, y_1)$ that the line passes through.

$$y - y_1 = m(x - x_1)$$

Step-by-Step Using Point-Gradient Form:

Determine the gradient $m$. (You might need to use two given points or the parallel/perpendicular rule.)
Substitute $m$ and the coordinates of one point $(x_1, y_1)$ into the formula.
Rearrange the equation into the required form (usually $y = mx + c$ or $ax + by + c = 0$).

4.3 The General Form

In many A-Level questions, you will be asked to present the answer in the general form.

$$ax + by + c = 0$$

Important Rule: $a$, $b$, and $c$ must be integers (whole numbers) and $a$ is usually kept positive.

Example: If you find $y = 0.5x - 3$, to convert it:
1. Multiply by 2: $2y = x - 6$
2. Rearrange to set equal to zero: $0 = x - 2y - 6$
(Here, $a=1, b=-2, c=-6$)

Key Takeaway: The Point-Gradient form $y - y_1 = m(x - x_1)$ is your best friend. Use it, then rearrange.

5. Intersections and Applications

Coordinate geometry often involves finding specific points or using the equations to solve geometric problems.

5.1 Finding the Intersection of Two Lines

When two lines intersect, they share a single common point $(x, y)$. This point satisfies both of their equations simultaneously.

Process:

Ensure both lines are written as equations (e.g., $L_1$ and $L_2$).
Treat the two equations as simultaneous equations.
Use substitution or elimination to solve for $x$.
Substitute your $x$ value back into one of the original equations to find $y$.
State the answer as a coordinate pair $(x, y)$.

Did you know? If the lines are parallel, they will never intersect, and when you try to solve the simultaneous equations, you will end up with a false statement (e.g., $0 = 5$).

5.2 Applications to Geometry

You can apply these formulas to various geometric shapes (triangles, squares, rhombuses, etc.). Key skills include:

Proving a right angle: Calculate the gradients of the two sides meeting at the vertex. If they are perpendicular (their product is -1), it's a right angle.
Proving a shape is a rhombus: Show that all four sides have equal length (using the distance formula).
Finding the perpendicular bisector: This is a common exam requirement.
1. Find the midpoint $M$ of the line segment.
2. Find the gradient $m_1$ of the line segment.
3. Calculate the perpendicular gradient $m_2$ (negative reciprocal).
4. Use the Point-Gradient form, substituting $M$ as the point $(x_1, y_1)$ and $m_2$ as the gradient.

Key Takeaway: Coordinate geometry links geometric properties (like right angles or parallel sides) directly to algebraic properties (like $m_1 m_2 = -1$ or $m_1 = m_2$).

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the formula for the distance $d$ between two points $A(x_1, y_1)$ and $B(x_2, y_2)$?

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

How do you find the midpoint $M$ of a line segment connecting two points?

Average the coordinates: $M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.

What is the formula for the gradient $m$ of a line passing through two points?

$m = \frac{y_2 - y_1}{x_2 - x_1}$ (also known as "Rise over Run").

What are the gradients of horizontal and vertical lines?

A horizontal line has a gradient of $0$, and a vertical line has an undefined gradient.

What condition must be met for two lines to be parallel?

Two lines are parallel if they have the same gradient: $m_1 = m_2$.

What is the relationship between the gradients of two perpendicular lines?

Their product is $-1$ ($m_1 \times m_2 = -1$), meaning one is the negative reciprocal of the other.

What is the gradient-intercept form of a straight line equation?

$y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept.

What is the point-gradient formula for finding the equation of a line?

$y - y_1 = m(x - x_1)$, where $m$ is the gradient and $(x_1, y_1)$ is a point on the line.

What is the general form of a line equation and what are its standard constraints?

$ax + by + c = 0$, where $a$, $b$, and $c$ must be integers.

How do you find the coordinates of the point where two lines intersect?

Solve the equations of the two lines as simultaneous equations.

What are the steps to find the equation of a perpendicular bisector?

Find the midpoint, find the negative reciprocal gradient, and use the point-gradient form.

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