[AQA] (A Level) Mathematics - C1 - Coordinate Geometry of Linear Equations

In order to find the distance between two points plotted on a set of axes, we may apply Pythagoras' theorem in terms of x and y:

    a^2 + b^2 = c^2 (where c is the hypotenuse)

By connecting the x coordinates of each point, a horizontal line is formed. Connecting the y coordinates forms a vertical line.

Together with the original line, this forms a right angle triangle, to which Pythagoras' theorem can be applied to find the length of 'c':

    sqrt([x2-x1]^2 + [y2-y1]^2) = distance (hypotenuse)

    e.g. Find the distance between A(1, 2) and B(7, 4)

sqrt([7-1]^2+[4-2]^2) = sqrt(6^2+2^2) = sqrt(36+4) = sqrt(40) = 6.32              

To find the midpoint of a line segment, we must find the values that fall in the exact centre of both the x and y coordinates of each point.

This means that the midpoint of the line segment itself lies precisely between the points in two dimensions, calculated as follows:

Finding the value in the centre of two numbers can be done by calculating the sum of the numbers and dividing them by two.

Therefore, it follows that the midpoint of a line segment joined by two points will have an x value of sum(x)/2 and a y value of sum(y)/2:

    Midpoint = ([x1 + x2]/2, [y1 + y2]/2)

    e.g. Find the midpoint of the line AB where A(2, 1) and B(8, 9)

M = ([2+8]/2, [1+9]/2) = (10/2, 10/2) = (5, 5)