Coordinate GeometryCambridge IGCSE Maths: Revision notes

The Cartesian coordinate system describes the position of a point in two dimensions using an ordered pair $(x, y)$.

• The $x$-coordinate gives the horizontal position
• The $y$-coordinate gives the vertical position
• Coordinates are always written in the order $(x, y)$

Understanding coordinates lets us describe and calculate properties of lines and shapes algebraically, without needing to see a picture.

The length of a line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is found using a form of Pythagoras' theorem: $\text{length} = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

This works because the horizontal and vertical distances between the two points form the two shorter sides of a right-angled triangle, and the line segment itself is the hypotenuse.

Example: Find the distance between $(1, 2)$ and $(4, 6)$. $\text{length} = \sqrt{(4-1)^2+(6-2)^2} = \sqrt{9+16} = \sqrt{25} = 5$

The midpoint of a line segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is the point exactly halfway between them: $\left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$

This is simply the average of the two $x$-coordinates and the average of the two $y$-coordinates.

Example: Find the midpoint of $(2, -3)$ and $(8, 5)$. $\left(\frac{2+8}{2}, \frac{-3+5}{2}\right) = (5, 1)$

Because Cambridge exam questions never rely on a drawn diagram, coordinate geometry problems must be solved purely by calculation.

• To find a distance: use the distance formula
• To find a midpoint: use the midpoint formula
• To check whether three points are collinear (lie on a straight line): calculate the gradient between each pair of points and confirm they are equal
• To find a missing coordinate given a midpoint: rearrange the midpoint formula for the unknown

Example: $M(4, 1)$ is the midpoint of $A(2, -3)$ and $B$. Find $B$. Since $\frac{2+x_B}{2}=4$, $x_B = 6$; since $\frac{-3+y_B}{2}=1$, $y_B=5$. So $B=(6,5)$.