The midpoint and the section formula

The midpoint

The midpoint of a line segment is the point halfway between its ends. Its coordinates are the averages of the coordinates of the ends. This is what you do in daily life: halfway between the numbers $2$ and $8$ is $5$, which is $\dfrac{2 + 8}{2}$.

For $A(x_1, y_1)$ and $B(x_2, y_2)$:

Example: the midpoint of $A(-2, 3)$ and $B(6, -5)$ is $\left(\dfrac{-2 + 6}{2}, \dfrac{3 + (-5)}{2}\right) = (2, -1)$.

If you know the midpoint and one end, you can find the other end. If $M(3, 4)$ is the midpoint of $A(1, 2)$ and $B$, then $\dfrac{1 + x}{2} = 3$ gives $x = 5$, and $\dfrac{2 + y}{2} = 4$ gives $y = 6$. So $B = (5, 6)$.

Dividing a line in a ratio

A point $P$ on the segment $AB$ can divide it in any ratio, not only $1 : 1$. We say that $P$ divides $AB$ internally in the ratio $m : n$ if

The section formula

Notice the pattern: $m$ goes with the far end $B$, and $n$ goes with the near end $A$. Check it: when $m = n$, the formula gives the midpoint.

Why does the formula work? Along the $x$-direction, the line goes from $x_1$ to $x_2$, a distance of $x_2 - x_1$. The point $P$ is $\dfrac{m}{m + n}$ of the way along. So its $x$-coordinate is $x_1 + \dfrac{m}{m + n}(x_2 - x_1) = \dfrac{n x_1 + m x_2}{m + n}$. The same holds for $y$. Parallel lines cut the segments in the same ratio, so the ratio is the same in both directions.

Worked example

Find the point $P$ that divides $A(1, 2)$ and $B(7, 5)$ in the ratio $1 : 2$.

Here $m = 1$ and $n = 2$.

• $x = \dfrac{1 \times 7 + 2 \times 1}{1 + 2} = \dfrac{9}{3} = 3$
• $y = \dfrac{1 \times 5 + 2 \times 2}{1 + 2} = \dfrac{9}{3} = 3$

So $P = (3, 3)$.

Check. $AP$ goes $2$ across and $1$ up. $PB$ goes $4$ across and $2$ up. $PB$ is twice $AP$, so the ratio is $1 : 2$. ✓

Set $m = n$. Whatever the two numbers are, $P$ lands on the midpoint, $(4, 3.5)$.

Finding the ratio

Example. In what ratio does the $x$-axis divide the line joining $A(2, -3)$ and $B(5, 6)$?

On the $x$-axis, $y = 0$. Let the ratio be $k : 1$. Then $y = \dfrac{6k + (-3)}{k + 1} = 0$, so $6k = 3$ and $k = \dfrac{1}{2}$. The ratio is $\dfrac{1}{2} : 1$, which is $1 : 2$. The point is $x = \dfrac{5 \times 1 + 2 \times 2}{3} = 3$, so $(3, 0)$.

Points of trisection

Two points that cut a line into three equal parts are the points of trisection. The one near $A$ divides $AB$ in the ratio $1 : 2$. The one near $B$ divides it in the ratio $2 : 1$.

Practise with new numbers

Try it in Python

This program uses the section formula.

Common mistakes

• Putting $m$ with the near end. In the formula, $m$ multiplies the coordinates of $B$.
• Adding and not averaging for the midpoint. Divide by $2$.
• Mixing up the ratio. $AP : PB = 1 : 2$ means that $P$ is nearer to $A$.
• Forgetting to divide by $m + n$.

Remember these

• The midpoint is the average of the coordinates.
• For $AP : PB = m : n$, the formula has $m$ with $B$ and $n$ with $A$.
• Trisection points are the ratios $1 : 2$ and $2 : 1$.
• To find the ratio in which an axis cuts a line, put the ratio as $k : 1$ and set a coordinate to $0$.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. What is the midpoint of $(2, 6)$ and $(8, -2)$?

2. The midpoint of $AB$ is $(4, 1)$, and $A = (0, -3)$. What is $B$?

3. Find the point that divides the line from $(0, 0)$ to $(10, 5)$ in the ratio $3 : 2$.

4. Find the points of trisection of the line from $(0, 0)$ to $(9, 6)$.