Dividing a line segment in a given ratio

This method finds the coordinates of the point $P$ that divides a line segment $AB$ in a given ratio $m:n$. It uses the section formula, which is a weighted average of the two endpoints.

It answers questions such as 'point $P$ divides $AB$ in the ratio $2:1$; find its coordinates', or 'the point $(5, 4)$ divides $AB$ in the ratio $2:1$, find $B$'. The midpoint formula is just the special case where the ratio is $1:1$, so this one method covers midpoints too.

It is a small, high-frequency skill that appears both on its own and inside longer Coordinate Geometry problems about parallelograms, medians and points of trisection.

Reach for this method whenever a question mentions a point dividing a segment 'in the ratio $m:n$', or uses phrases like 'internal point of division', 'trisects $AB$', or 'divides $AB$ such that $AP:PB = 2:3$'. If the ratio is $1:1$, the point is the midpoint and the same formula applies.

You can also run the formula backwards. If the dividing point and one endpoint are given along with the ratio, substitute what you know and solve for the missing endpoint's coordinates.

The key first move is always to read the ratio in the correct order, $AP:PB$, measured from the first named point $A$, because getting $m$ and $n$ the wrong way round moves $P$ to the wrong place.

1. 1

   Label the endpoints

   Set $A(x_1, y_1)$ as the first point and $B(x_2, y_2)$ as the second, in the order the ratio is measured.

2. 2

   Read the ratio as m : n

   For $AP:PB = m:n$, $m$ is the part next to $A$ and $n$ is the part next to $B$.

3. 3

   Write the section formula

   $P = \left(\frac{n x_1 + m x_2}{m + n}, \frac{n y_1 + m y_2}{m + n}\right)$: the first point is weighted by $n$, the second by $m$.

4. 4

   Substitute the numbers

   Put in the coordinates and the ratio values, keeping the denominator $m + n$.

5. 5

   Simplify each coordinate

   Work out the $x$-coordinate and the $y$-coordinate separately and state $P$.

6. 6

   Sanity-check the position

   Confirm $P$ lies between $A$ and $B$ and is nearer the endpoint the ratio says it should be.

• Swapping $m$ and $n$ in the formula, remember the first point $A$ is weighted by $n$, and the second point $B$ by $m$.
• Reading the ratio the wrong way round: $AP:PB = 2:1$ is not the same as $PB:AP = 2:1$.
• Writing the denominator as $m$ or $n$ alone instead of $m + n$.
• Forgetting that the midpoint is simply the ratio $1:1$, then using a different (wrong) formula for it.
• Sign slips when an endpoint has a negative coordinate, especially in the numerator.

Almost every lost mark here comes from one place: pairing the ratio numbers with the wrong points. Students instinctively multiply $A$ by $m$ because $m$ is written first, when the formula actually weights $A$ by $n$.

In one-to-one lessons our teachers give you a reliable habit, write $A(x_1, y_1)$, $B(x_2, y_2)$ and $AP:PB = m:n$ in a fixed layout every time, then read the weights straight off. We also make you do the quick 'fraction of the way' sanity check you saw above, which catches a swapped ratio in seconds.

Because SPM Add Math uses analytic marking, a correctly substituted formula earns method marks even if the final arithmetic slips. Every teacher on spmaddmath.com.my is experienced.

Lessons are online and taught in English, at your own pace.