Coordinate geometry in the (x, y) plane

Q: What formula is used to find the distance between two points $ (x_1, y_1) $ and $ (x_2, y_2) $?

The Distance Formula is $ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $.

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

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

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

The gradient is calculated as $ m = \frac{y_2 - y_1}{x_2 - x_1} $.

Q: What does a positive vs. a negative gradient indicate about the direction of a line?

A positive gradient slopes upwards from left to right, while a negative gradient slopes downwards.

Q: What is the rule for the gradients of two parallel lines?

If two lines are parallel, their gradients are equal: $ m_1 = m_2 $.

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

The gradients are negative reciprocals , meaning $ m_1 \times m_2 = -1 $ (or $ m_2 = -\frac{1}{m_1} $).

Q: In the equation $ y = mx + c $, what do the letters $ m $ and $ c $ represent?

$ m $ is the gradient and $ c $ is the $ y $-intercept .

Q: What is the Point-Gradient formula for the equation of a straight line?

The formula is $ y - y_1 = m(x - x_1) $, where $ m $ is the gradient and $ (x_1, y_1) $ is a known point.

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

Solve the equations of the two lines as simultaneous equations .

Mathematics (YMA01) · Pearson Edexcel International A Level · 6 min read · Updated: Sep 7, 2026

Welcome to Coordinate Geometry in the (x, y) Plane!

Hello there! This chapter, Coordinate Geometry in the (x, y) Plane, is fundamental to Pure Mathematics. It’s where we use algebra (equations) to describe geometry (shapes, lines, and positions).

Think of the coordinate plane as a massive map. If you know the rules, you can calculate the distance between any two locations, figure out the steepest path, and describe exactly where a route is going.
Mastering this topic will set you up for success in later chapters like Calculus and Vectors, where these coordinates become crucial. Don't worry if some concepts look familiar from previous studies; we will build on them rigorously!

Section 1: Locating Points and Calculating Distances

1.1 Understanding Coordinates and the Cartesian Plane

A point on the plane is represented by a pair of coordinates, $(x, y)$. The $x$-coordinate tells you the horizontal position, and the $y$-coordinate tells you the vertical position.

Quick Review: Prerequisite Concepts

The origin is the point $(0, 0)$.
Moving right increases $x$. Moving left decreases $x$.
Moving up increases $y$. Moving down decreases $y$.

1.2 The Distance Between Two Points

How far is point A from point B? We can find the length of the line segment connecting them using the Distance Formula.

This formula is simply a disguised version of the Pythagoras’ Theorem ($a^2 + b^2 = c^2$). If you draw a right-angled triangle between the two points, the distance is the hypotenuse!

Step-by-Step: The Distance Formula

Let the two points be $A(x_1, y_1)$ and $B(x_2, y_2)$.

The distance $d$ is calculated as:
$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$$

Analogy: Imagine calculating the distance you walk (the change in x) and the height you climb (the change in y). You square both changes, add them, and then take the square root to find the direct, straight-line distance.

Important Tip: Because you are squaring the differences, it doesn't matter which point you label $(x_1, y_1)$ and which you label $(x_2, y_2)$. The final answer will be the same, so long as you are consistent within each bracket.

1.3 Finding the Midpoint

The Midpoint is the exact center point of the line segment joining two points. To find the midpoint, we simply calculate the average of the $x$-coordinates and the average of the $y$-coordinates.

The Midpoint Formula

For points $A(x_1, y_1)$ and $B(x_2, y_2)$, the midpoint $M$ is:
$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Key Takeaway: Distance uses the square root and difference (Pythagoras). Midpoint uses averages and addition.

Section 2: Understanding Steepness – The Gradient ($m$)

The gradient (often represented by the letter $m$) is a measure of the steepness and direction of a line.

2.1 Calculating the Gradient

The gradient is defined as the change in the vertical direction (rise) divided by the change in the horizontal direction (run).

The Gradient Formula

For points $A(x_1, y_1)$ and $B(x_2, y_2)$:
$$m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1}$$

Did you know?

The letter m is thought to have come from the French word monter, meaning "to climb" or "to rise."

Interpreting the Gradient

Positive Gradient ($m > 0$): The line slopes upwards from left to right.
Negative Gradient ($m < 0$): The line slopes downwards from left to right.
Zero Gradient ($m = 0$): The line is perfectly horizontal (e.g., $y = 5$).
Undefined Gradient: The line is perfectly vertical (e.g., $x = 3$). (You cannot divide by zero in the formula!)

2.2 Parallel and Perpendicular Lines

This section is essential for proving geometric relationships!

Parallel Lines

If two lines are parallel, they are running in the exact same direction and will never intersect.

Rule 1: If line $L_1$ is parallel to line $L_2$, then their gradients are equal.
$$m_1 = m_2$$

Perpendicular Lines

If two lines are perpendicular, they meet at a right angle ($90^\circ$).

Rule 2: If line $L_1$ is perpendicular to line $L_2$, their gradients are negative reciprocals of each other.

This means:
$$m_1 \times m_2 = -1 \quad \text{or} \quad m_2 = -\frac{1}{m_1}$$

Memory Aid: To find the perpendicular gradient, flip the fraction and change the sign!
Example: If $m_1 = \frac{2}{3}$, then $m_2 = -\frac{3}{2}$. If $m_1 = -4$, then $m_2 = \frac{1}{4}$.

Common Mistake to Avoid

Students often forget to change the sign when finding the perpendicular gradient. A perpendicular gradient must have the opposite sign (one positive, one negative).

Key Takeaway: Parallel lines have the same steepness. Perpendicular lines use the 'flip and negate' trick.

Section 3: Defining the Line – Equations of Straight Lines

The ultimate goal of coordinate geometry is to describe the entire line using a single algebraic equation. There are two primary forms you need to master.

3.1 The Gradient-Intercept Form

This is probably the most familiar form, often used when plotting graphs quickly.

$$y = mx + c$$

Where:

$m$ is the gradient (steepness).
$c$ is the $y$-intercept (the point where the line crosses the y-axis, i.e., where $x=0$).

3.2 The Point-Gradient Form (The Workhorse Formula)

While $y = mx + c$ is great for visualization, the Point-Gradient Form is usually the quickest way to find the equation of a line when you know the gradient ($m$) and any single point $(x_1, y_1)$ on the line.

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

Encouragement: This form might look a bit intimidating at first, but it is your best friend in A-Level Maths! It avoids finding $c$ separately, saving you time and avoiding calculation errors.

Step-by-Step: Using the Point-Gradient Form

Calculate $m$: Find the gradient using two points, or use the parallel/perpendicular rule if $m$ is given.
Choose a point $(x_1, y_1)$: Use one of the points given on the line.
Substitute: Plug $m$, $x_1$, and $y_1$ into the formula $y - y_1 = m(x - x_1)$.
Simplify: Rearrange the equation into the required format (usually $y = mx + c$ or $ax + by + c = 0$).

3.3 Intersection of Two Lines

When two straight lines cross, they share a single point $(x, y)$ called the point of intersection. At this point, the $x$ and $y$ values satisfy both equations simultaneously.

To find the intersection, you must solve the two equations as a pair of simultaneous equations.

Process of Finding Intersection

If you have:
Line 1: $y = m_1x + c_1$
Line 2: $y = m_2x + c_2$

Since $y$ is the same at the intersection, you can set the right-hand sides equal to each other (Substitution Method):
$$m_1x + c_1 = m_2x + c_2$$

Solve this resulting linear equation to find the value of $x$.
Substitute the value of $x$ back into either of the original line equations to find the value of $y$.
State the answer as a coordinate $(x, y)$.

Key Takeaway: The Point-Gradient form is the most efficient method for finding the equation of a line. Intersection points are found by solving simultaneous equations.

Quick Review Summary

Coordinate Geometry Toolkit

Distance: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ (Pythagoras)
Midpoint: $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ (Averages)
Gradient: $m = \frac{y_2 - y_1}{x_2 - x_1}$ (Rise over Run)
Parallel Lines: $m_1 = m_2$
Perpendicular Lines: $m_1 \times m_2 = -1$ (Negative Reciprocal)
Equation (Best form): $y - y_1 = m(x - x_1)$

Keep practicing these formulas until they are second nature. You’ve got this!

Quick check

Can you answer these now?

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

What formula is used to find the distance between two points $ (x_1, y_1) $ and $ (x_2, y_2) $?

The Distance Formula is $ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $.

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

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

What is the formula for calculating the gradient $ m $ of a line between two points?

The gradient is calculated as $ m = \frac{y_2 - y_1}{x_2 - x_1} $.

What does a positive vs. a negative gradient indicate about the direction of a line?

A positive gradient slopes upwards from left to right, while a negative gradient slopes downwards.

What is the rule for the gradients of two parallel lines?

If two lines are parallel, their gradients are equal: $ m_1 = m_2 $.

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

The gradients are negative reciprocals, meaning $ m_1 \times m_2 = -1 $ (or $ m_2 = -\frac{1}{m_1} $).

In the equation $ y = mx + c $, what do the letters $ m $ and $ c $ represent?

$ m $ is the gradient and $ c $ is the $ y $-intercept.

What is the Point-Gradient formula for the equation of a straight line?

The formula is $ y - y_1 = m(x - x_1) $, where $ m $ is the gradient and $ (x_1, y_1) $ is a known point.

How do you find the point where two straight lines intersect?

Solve the equations of the two lines as simultaneous equations.

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