2.12.1 Midpoint of a line

The study of geometry using coordinate points is known as coordinate geometry.

If you are given two points at GCSE, you must be able to find

A point that is precisely halfway between the two points is known as the midpoint of a line segment.

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

\( A \) is the point \( (2, 3) \) and \( B \) is the point \( (7,-5) \).

Find the coordinates of the midpoint of \( AB \).

The midpoint of \( ( x_1 , y_1 ) \) and \( (x_2 , y_2 ) \) is \( (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}) \)

Midpoint of \( AB= (\frac{2+7}{2} , \frac{3+(-7)}{2}) \)

\( =(\frac{9}{2}, -\frac{4}{2}) \)

\( =(4.5,-2) \)

\( (4.5,-2)\)

To extend the idea of the midpoint

As midpoint splits the line \( AB \) in the ratio \( 1 \) : \( 1 \)
If you are asked to divide the line \( AB \) in the ratio \( m \) : \( n \) then you need to find the point that lies \( \frac {m}{m+n} \) of the way from \( A \) to \( B \)
If you are asked to divide the line \( AB \) in the ratio \( 1 \) : \( n \)

Find the difference between the \( x \) coordinates
Divide the difference by \( (1+n) \), add the result to the \( x \) coordinate of \( A \)
Repeat for the \( y \) coordinates

\( A \) is the point \( (-4,3) \) and \( B \) is the point \( (8,-12) \).

\( A \) point \( N \) divides \( AB \) in the ratio \( 1 \) : \( 3 \) Find the coordinates of \( N \).

Find the difference between \( x \) coordinates

\( (8-(-4))=12 \)

Divide the difference by \( 1+3=4 \)

\( 12\div 4=3 \)

Add this to the \( x\) coordinate of \( A\)

\( -4+3=-1 \)

\( x \) coordinate of \( N \) is \( -1 \)

Now find the difference between \( y \) coordinates

\( -12-3=-15 \)

Divide the difference by \( 1+3=4 \)

\( -15\div 4=-3.75 \)

Add this to the \( y \) coordinate of \( A \)

\( 3+(-3.75)=-0.75 \)

\( y \) coordinate of \( N \) is \( -0.75 \)

So, the coordinates of the point \( N \) are \( (-1, -0.75) \)