Points, Distance and Midpoint

Introduction

Any point can be described by a pair of numbers \((x, y)\). The numbers are the distance of A from the y axis \((x)\) , and from the x axis \((y)\). The pair \((x, y)\) are called the coordinates of the point A. Any points to the left of the y-axis will have a negative x coordinate. Any points below the x axis, will have a negative y coordinate.

The distance between 2 points

Find the distance between two points \(A (1,2)\) and \(B (4,5)\).

To find the length \(AB\) we use Pythagoras’ Theorem. \(AB\) is the hypotenuse of an appropriate right-angled triangle \(ABC\). This means that \(C\) must be the point \((4,2)\).

The distance \(AC\) is \(4 − 1 = 3\)

The distance \(BC\) is \(5 − 2 = 3\)

According to Pythagoras Theorem

\(AB^2 = AC^2 + BC^2\)

Substitute the values for \(AC\) and \(BC\)

\(AB^2 = 3^2 + 3^2\)

\(AB^2 = 9 + 9 =18\)

\(AB = \sqrt{18} = 4.243\)

The distance between \(A\) and \(B\) is \(4.243\).

You can derive a general formula to use instead

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

More examples you can find here

Find the midpoint of two points\(A (2, 3)\) and \(B (4, 5)\).

Let \(C\) be the midpoint of the line segment. To find the coordinates of \(C\) the \(x\) coordinate must be the average of the \(x\) coordinates of \(A\) and \(B\).

The \(y\) coordinate must be the average of the \(y\) coordinates of \(A\) and \(B\).

The \(x\) coordinate is \(\displaystyle \frac{1}{2} (2 + 4) = 3\).

The \(y\) coordinate is \(\displaystyle \frac{1}{2} (3 + 5) = 4\).

So \(C\) has coordinates \((3, 4)\)

Now we can derive a general formula for the midpoint. If the two points are \(A (x1, y1)\) and \(B (x2, y2)\) then the midpoint \(C\) must be equal to

\(\displaystyle \frac{1}{2} (x_1 + x_2),\frac{1}{2} (y_1 + y_2)\)