Welcome to Coordinate Geometry in the \((x, y)\) Plane!
Hello! Coordinate geometry is one of the most fundamental and practical areas of Pure Mathematics. If you’ve ever used a map, GPS, or even played a game on a grid, you’ve used coordinates!
In this chapter, we move beyond just plotting points. We will learn how to measure the distance between points, find the exact middle, determine the steepness of a line, and write the algebraic rule (the equation) that defines that line.
Why is this important? This unit provides the foundational skills needed for almost all geometric problems in advanced mathematics, allowing us to translate visual shapes into solvable algebraic equations. Don't worry if some concepts seem unfamiliar; we will break everything down step-by-step!
Section 1: The Basics of the Cartesian Plane
What is a Coordinate?
A coordinate specifies the position of a point using an ordered pair, usually written as \((x, y)\).
- The \(x\)-coordinate (the first number) tells you how far to move horizontally (left or right).
- The \(y\)-coordinate (the second number) tells you how far to move vertically (up or down).
Did you know?
The coordinate system is named after the French mathematician and philosopher René Descartes. Legend says he invented it while lying in bed, watching a fly crawl on the ceiling, realizing he could describe its exact location using perpendicular measurements!
Key Takeaway: Coordinates are simply addresses on a two-dimensional map.
Section 2: Measuring Segments – Distance and Midpoint
We often need to find the physical properties of a line segment connecting two points, \(A(x_1, y_1)\) and \(B(x_2, y_2)\).
1. The Distance Between Two Points
Finding the distance is like using the Pythagorean Theorem. We imagine the line segment as the hypotenuse of a right-angled triangle.
The length of the horizontal side is the difference in the \(x\) coordinates (\(x_2 - x_1\)). The length of the vertical side is the difference in the \(y\) coordinates (\(y_2 - y_1\)).
The Distance Formula (\(d\))
The distance \(d\) between points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]
Step-by-Step Example: Find the distance between \((1, 5)\) and \((4, 9)\).
1. Identify coordinates: \(x_1=1, y_1=5\) and \(x_2=4, y_2=9\).
2. Find the differences: \((4 - 1) = 3\) and \((9 - 5) = 4\).
3. Square and add: \(3^2 + 4^2 = 9 + 16 = 25\).
4. Take the square root: \(\sqrt{25} = 5\).
The distance is 5 units.
2. The Midpoint of a Line Segment
The midpoint is the exact center of the line segment. To find the center, we simply calculate the average of the \(x\) coordinates and the average of the \(y\) coordinates separately.
The Midpoint Formula (\(M\))
The midpoint \(M\) between points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
Common Mistake to Avoid: The distance formula involves subtraction and squaring. The midpoint formula involves addition and division by 2. Don't mix them up!
Key Takeaway: Distance uses Pythagoras (differences squared); Midpoint uses Averages (sums divided by 2).
Section 3: Understanding Steepness – The Gradient (\(m\))
The gradient (\(m\)) measures the steepness and direction of a line. Think of it as the slope of a hill.
1. Calculating the Gradient
We calculate the gradient by comparing the vertical change (the 'rise') to the horizontal change (the 'run').
The Gradient Formula
For two points \((x_1, y_1)\) and \((x_2, y_2)\):
\[m = \frac{\text{Change in } y}{\text{Change in } x} = \frac{y_2 - y_1}{x_2 - x_1}\]
Memory Aid: Gradient is Rise over Run.
2. Interpreting the Gradient
- Positive Gradient (\(m > 0\)): The line slopes upwards from left to right (an uphill climb).
- Negative Gradient (\(m < 0\)): The line slopes downwards from left to right (a downhill slide).
- Zero Gradient (\(m = 0\)): This is a horizontal line (a flat road). The numerator \((y_2 - y_1)\) is zero.
- Undefined Gradient: This is a vertical line. The denominator \((x_2 - x_1)\) is zero, and we cannot divide by zero.
Step-by-Step Example: Find the gradient between \((1, 8)\) and \((5, 0)\).
1. Rise (Change in \(y\)): \(0 - 8 = -8\).
2. Run (Change in \(x\)): \(5 - 1 = 4\).
3. Calculate \(m\): \(m = \frac{-8}{4} = -2\).
The line is relatively steep and slopes downwards.
Quick Review: The sign of the gradient tells you the direction, and the magnitude (the number) tells you the steepness. A gradient of 10 is much steeper than a gradient of 1.
Section 4: Relationships Between Lines – Parallel and Perpendicular
In geometry, lines often interact. We use gradients to determine if two lines are parallel or perpendicular.
1. Parallel Lines
Parallel lines are lines that run in the same direction and never intersect (like train tracks).
If line \(L_1\) has gradient \(m_1\) and line \(L_2\) has gradient \(m_2\), then:
\[\text{If } L_1 \text{ is parallel to } L_2 \text{, then } \mathbf{m_1 = m_2}\]
If they have the same steepness, they must be parallel!
2. Perpendicular Lines
Perpendicular lines intersect at a perfect right angle (\(90^\circ\)). This relationship is essential for many geometric proofs and calculations.
The Perpendicular Rule
If line \(L_1\) with gradient \(m_1\) is perpendicular to line \(L_2\) with gradient \(m_2\), then:
\[\mathbf{m_1 \times m_2 = -1} \quad \text{or} \quad \mathbf{m_2 = -\frac{1}{m_1}}\]
Memory Trick: To find the gradient of a perpendicular line, you must flip it and negate it (take the negative reciprocal).
Example: If \(m_1 = 3\), the perpendicular gradient \(m_2\) is \(-\frac{1}{3}\).
Example: If \(m_1 = -\frac{2}{5}\), the perpendicular gradient \(m_2\) is \(+\frac{5}{2}\).
Accessibility Note on Vertical/Horizontal Lines: A vertical line (undefined gradient) is always perpendicular to a horizontal line (gradient \(m=0\)). The formula \(m_1 m_2 = -1\) doesn't apply directly here, so remember this special case visually!
Key Takeaway: Parallel means Equal Gradients. Perpendicular means Negative Reciprocal Gradients.
Section 5: The Equation of a Straight Line
The equation of a line is the algebraic rule that every single point \((x, y)\) on that line must obey. We primarily use two forms in A Level Maths.
1. Slope-Intercept Form: \(y = mx + c\)
This is the most famous form and is great for plotting and quick analysis.
- \(m\) is the gradient (steepness).
- \(c\) is the \(y\)-intercept (the point where the line crosses the \(y\)-axis, i.e., where \(x=0\)).
2. Point-Gradient Form: \(y - y_1 = m(x - x_1)\)
This form is usually the quickest way to find the equation of a line when you know the gradient \(m\) and at least one point \((x_1, y_1)\) on the line.
You substitute your values into this form and then rearrange it into the standard \(y = mx + c\) or \(ax + by + c = 0\) (often required by the exam) at the end.
Step-by-Step Process: Finding the Equation
We want to find the equation of the line passing through points \(A(3, 10)\) and \(B(-1, 2)\).
-
Find the Gradient (\(m\)):
\[m = \frac{10 - 2}{3 - (-1)} = \frac{8}{4} = 2\] -
Use the Point-Gradient Form: Choose one point (e.g., \((3, 10)\)) and the gradient \(m=2\).
\[y - y_1 = m(x - x_1)\] \[y - 10 = 2(x - 3)\] -
Rearrange into \(y = mx + c\):
\[y - 10 = 2x - 6\] \[y = 2x + 4\]
The equation of the line is \(y = 2x + 4\).
Encouragement: Mastering the use of the Point-Gradient form is essential. It prevents calculation errors and is always the fastest starting point!
Section 6: Intersection of Lines and Final Review
Finding the Point of Intersection
When two straight lines cross, the point where they meet is called the point of intersection. At this specific point, the \(x\) and \(y\) coordinates satisfy both line equations simultaneously.
To find the intersection, you simply solve the two line equations as a pair of simultaneous equations.
Example Process:
Find the intersection of Line 1: \(y = 3x - 5\) and Line 2: \(2x + y = 10\).
-
Substitute: Since Line 1 is already solved for \(y\), substitute \((3x - 5)\) into the \(y\) variable in Line 2.
\[2x + (3x - 5) = 10\] -
Solve for \(x\):
\[5x - 5 = 10\] \[5x = 15\] \[x = 3\] -
Find \(y\): Substitute \(x=3\) back into either original equation (Line 1 is easiest).
\[y = 3(3) - 5\] \[y = 9 - 5 = 4\]
The lines intersect at the point \((3, 4)\).
Quick Review Checklist for P1 Coordinate Geometry
The Coordinate Geometry skills required for P1 are all interconnected. If you can confidently answer these four questions, you are ready!
| Concept | Formula/Rule | Purpose |
|---|---|---|
| Distance | \(d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) | Finding the length of a line segment. |
| Midpoint | \(M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\) | Finding the exact center of a segment. |
| Gradient | \(m = \frac{y_2 - y_1}{x_2 - x_1}\) | Determining steepness and direction (Rise/Run). |
| Perpendicular Lines | \(m_1 = -1/m_2\) | Identifying a right-angle relationship. |
| Equation of Line | \(y - y_1 = m(x - x_1)\) | Writing the algebraic rule for the line. |
Final Encouragement: Coordinate geometry is about translating shapes into numbers. Practice makes perfect, especially with calculating gradients and using the negative reciprocal rule! You've got this!