Coordinates - Mathematics (0580) - Cambridge IGCSE

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Mathematics (0580) Study Notes: Coordinate Geometry

Welcome to the Coordinate Geometry chapter! This topic is all about linking Algebra and Geometry by using numbers (coordinates) to describe exact locations in a 2D space. Mastering these tools will allow you to find distances, calculate steepness, and describe lines using simple equations. It’s essential for understanding graphs, physics, and map reading!

1. The Basics: Cartesian Coordinates (C4.1 / E4.1)

What are Cartesian Coordinates?

The Cartesian coordinate system uses two axes—the horizontal axis ($x$-axis) and the vertical axis ($y$-axis)—which intersect at the origin, $(0, 0)$. We use an ordered pair $(x, y)$ to define the exact location of a point.

The $x$-coordinate is the distance along the horizontal axis.
The $y$-coordinate is the distance along the vertical axis.

Memory Aid: The Airplane Rule

Think of plotting a point like taking off from an airport:

Walk Across ($x$) before you Climb Up/Down ($y$). Always state the $x$ value first, then the $y$ value: $(x, y)$.

Example: The point $(3, -5)$ is 3 units right of the origin and 5 units down.

Quick Review: Coordinates

A coordinate pair is always written $(x, y)$. The axes divide the plane into four regions called quadrants.

2. Gradient of a Straight Line (C4.2 / E4.2)

What is Gradient?

The gradient ($m$) of a straight line measures its steepness and direction. Think of it as the slope of a roof or a hill.

We calculate the gradient using the ratio of the vertical change (the rise) to the horizontal change (the run).

$$m = \frac{\text{Vertical Change (Rise)}}{\text{Horizontal Change (Run)}}$$

Finding Gradient from Two Points (E4.2)

If you have two points, $(x_1, y_1)$ and $(x_2, y_2)$, the formula for the gradient is:

$$m = \frac{y_2 - y_1}{x_2 - x_1}$$

Did you know? It doesn't matter which point you label as $(x_1, y_1)$ and which as $(x_2, y_2)$, as long as you are consistent across the numerator and the denominator!

Types of Gradient

Positive Gradient ($m > 0$): The line slopes upwards from left to right (like climbing a hill).
Negative Gradient ($m < 0$): The line slopes downwards from left to right (like skiing downhill).
Zero Gradient ($m = 0$): A horizontal line (flat ground). This line has the equation $y = k$.
Undefined Gradient: A vertical line (a cliff face). This line has the equation $x = k$.

Key Takeaway: Gradient

Gradient ($m$) is Rise over Run. Consistency is key when using the formula: subtract the coordinates in the same order.

3. Distance and Midpoint (C4.3 / E4.3)

We can use coordinates to find the physical distance between two points and the exact centre point of the line segment connecting them.

Finding the Midpoint (C4.3 / E4.3)

The midpoint ($M$) is simply the average of the $x$-coordinates and the average of the $y$-coordinates.

If the end points are $(x_1, y_1)$ and $(x_2, y_2)$, the midpoint $M$ is:

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

Example: Find the midpoint of $A(2, 8)$ and $B(6, 4)$.
$$M = \left( \frac{2 + 6}{2}, \frac{8 + 4}{2} \right) = \left( \frac{8}{2}, \frac{12}{2} \right) = (4, 6)$$

Calculating the Length (Distance) (C4.3 / E4.3)

To find the length ($D$) of the line segment, we use Pythagoras' Theorem ($a^2 + b^2 = c^2$). Imagine the line segment is the hypotenuse of a right-angled triangle. The distance formula is:

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

Step-by-Step Distance Calculation:

Find the difference in $x$ values: $(x_2 - x_1)$.
Find the difference in $y$ values: $(y_2 - y_1)$.
Square both differences.
Add the squared differences together.
Take the square root of the result.

Key Takeaway: Midpoint and Distance

Midpoint uses addition (averaging). Distance uses subtraction and squares (like Pythagoras).

4. Equations of Linear Graphs (C4.4 / E4.4)

The Gradient-Intercept Form

The equation of a straight line is most commonly written in the gradient-intercept form:

$$y = mx + c$$

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

Example: For the line $y = 6x + 3$, the gradient is $m=6$, and the line crosses the $y$-axis at $(0, 3)$.

Obtaining the Equation of a Line

If you are given the gradient ($m$) and a point $(x_1, y_1)$ on the line, you can find $c$:

Step 1: Find $m$ (use the gradient formula if given two points).

Step 2: Substitute $m$, $x_1$, and $y_1$ into the equation $y = mx + c$.

Step 3: Solve the resulting equation to find $c$.

Step 4: Write the final equation using $m$ and $c$.

Example: A line has a gradient $m = 4$ and passes through $(1, -3)$.
Substitute into $y = mx + c$:
$-3 = 4(1) + c$
$-3 = 4 + c$
$c = -7$
The equation is: $y = 4x - 7$.

Special Lines (C4.4 / E4.4)

The syllabus requires you to recognize the equations of horizontal and vertical lines:

Horizontal Line: $y = k$. The gradient is zero. (e.g., $y = 5$)
Vertical Line: $x = k$. The gradient is undefined. (e.g., $x = -2$)

Key Takeaway: Linear Equations

Every non-vertical straight line can be defined by its steepness ($m$) and its starting point ($c$) using $y = mx + c$. Always ensure your final equation is in a fully simplified form.

5. Relationships Between Lines

Parallel Lines (C4.5 / E4.5)

Parallel lines are lines that run next to each other and never intersect. This means they must have the exact same steepness.

If line 1 has gradient $m_1$ and line 2 has gradient $m_2$, then for them to be parallel:

$$m_1 = m_2$$

Example: Find the equation of the line parallel to $y = 4x - 1$ that passes through $(1, -3)$.
The gradient of the given line is $m = 4$. The parallel line must also have $m = 4$.
Use $m=4$ and point $(1, -3)$ to find $c$:
$-3 = 4(1) + c$
$c = -7$
Equation: $y = 4x - 7$.

Perpendicular Lines (E4.6 - Extended Content Only)

Perpendicular lines meet at a right angle ($90^\circ$). Their gradients are related in a special way: they are negative reciprocals of each other.

If line 1 has gradient $m_1$, the gradient of the perpendicular line, $m_2$, is found by:

$$m_2 = -\frac{1}{m_1}$$

$$m_1 \times m_2 = -1$$

Memory Trick: "Flip it and Change the Sign"

If $m_1 = 2$, then $m_2 = -\frac{1}{2}$
If $m_1 = -\frac{3}{4}$, then $m_2 = +\frac{4}{3}$
If $m_1 = -5$, then $m_2 = \frac{1}{5}$

Step-by-Step Finding a Perpendicular Line:

Find the gradient ($m_1$) of the original line (rearrange to $y=mx+c$ if needed).
Calculate the negative reciprocal to find the new gradient ($m_2$).
Use the new gradient ($m_2$) and the given point $(x_1, y_1)$ to find the $y$-intercept $c$.
Write the final equation in the form $y = m_2x + c$.

Quick Review: Parallel and Perpendicular

Parallel: Same gradient ($m_1 = m_2$).
Perpendicular (Extended): Negative reciprocal gradient ($m_1 \times m_2 = -1$).