Find the gradient, length and midpoint of a line segment, and find the equation of a straight line through given points

What this dot point is asking

SEAB wants you to handle the basics of the straight line on the coordinate plane: find the gradient (steepness) of a segment, its length, and its midpoint, and write down the equation of a line given enough information. These are the building blocks for the parallel and perpendicular line work and the circle topic, and they reappear in calculus when you find tangents and normals.

The answer

Gradient

The gradient of the line through $(x_1, y_1)$ and $(x_2, y_2)$ measures how steep it is, as the change in $y$ divided by the change in $x$:

Keep the order of subtraction the same on top and bottom. A positive gradient slopes up to the right; a negative gradient slopes down; a horizontal line has gradient $0$.

Length of a segment

The length (distance) between the two points comes from Pythagoras:

Leave the answer as an exact surd unless a decimal is asked for.

Midpoint

The midpoint is the average of the two coordinates:

The equation of a line

Two convenient forms:

• Gradient-intercept: $y = mx + c$, where $m$ is the gradient and $c$ is the $y$-intercept.
• Point-gradient: $y - y_1 = m(x - x_1)$, used when you know the gradient and one point.

To find the line through two points, first compute the gradient, then substitute one point into the point-gradient form, and finally rearrange into the form requested.

Examples in context

Example 1. Perpendicular bisector. To find the perpendicular bisector of a segment you need both its midpoint and its gradient: the bisector passes through the midpoint with the perpendicular gradient. So these three segment formulas are the first step of a very common coordinate-geometry construction.

Example 2. Distance to check a shape. Given four points, computing the lengths of the sides with the distance formula shows whether a quadrilateral is a square, rectangle or rhombus. The length formula is the standard tool for verifying geometric properties from coordinates.

Try this

Q1. Find the gradient of the line through $(2, 1)$ and $(6, 9)$. [2 marks]

• Cue. $m = \dfrac{9 - 1}{6 - 2} = \dfrac{8}{4} = 2$.

Q2. Find the midpoint of the segment joining $(-2, 5)$ and $(4, 1)$. [2 marks]

• Cue. $\left(\dfrac{-2 + 4}{2}, \dfrac{5 + 1}{2}\right) = (1, 3)$.

Q3. Find the equation of the line with gradient $-3$ passing through $(1, 2)$, in the form $y = mx + c$. [2 marks]

• Cue. $y - 2 = -3(x - 1) \Rightarrow y = -3x + 5$.