Content description

Find the midpoint and gradient of a line segment (interval) on the Cartesian plane using a range of strategies, including graphing software (ACMNA294)

Source: Australian Curriculum, Assessment and Reporting Authority (ACARA)

The midpoint of an interval

The coordinates of the midpoint of a line interval can be found using averages as we will see.

We first deal with the situation where the points are horizontally or vertically aligned.

Example 1

Find the coordinates of the midpoint of the line interval AB, given the following points:

1. \(A(1, 2)\) and \(B(7, 2)\)
2. \(A(1, −2)\) and \(B(1, 3)\)

Solution

1. AB is a horizontal line interval, the midpoint is at (4, 2), since 4 is halfway between 1 and 7.
   
   Note that 4 is the average of 1 and 7, that is: 4 = \(\dfrac{1 + 7}{2}\).
2. The midpoint of AB has coordinates \((1, \dfrac{1}{2})\).
   
   Note that \(\dfrac{1}{2}\) is the average of 3 and \(-2\).