Section formula

Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ be two fixed points, and let $R(x,y)$ divide $PQ$ internally in the ratio $m:n$, meaning $R$ lies between $P$ and $Q$ with $PR:RQ = m:n$.

where: $m,n$ are the two parts of the ratio ($m$ is the share nearer $Q$, $n$ the share nearer $P$), and $x,y$ are the coordinates of the dividing point $R$. If $R$ divides $PQ$ externally in the ratio $m:n$ — so $R$ sits on the line through $P$ and $Q$ but outside the segment itself — every plus sign flips to a minus sign:

A handy way to hold both formulas in one head: external division is just internal division with $n$ replaced by $-n$. You never truly need two separate rules. Two special cases are worth naming on sight. Putting $m=n=1$ in the internal formula collapses it to the familiar mid-point formula,

and applying the section formula to a triangle's median (which a centroid divides $2:1$ from the vertex) gives the centroid,

for a triangle with vertices $A(x_1,y_1)$, $B(x_2,y_2)$, $C(x_3,y_3)$ — most students simply memorise this averaged form directly rather than re-deriving it each time. It helps to keep the two formulas visually distinct in memory. The internal-division formula always has $m+n$ sitting in the denominator with matching plus signs on top; the moment either endpoint's coefficient turns negative in the denominator, you have switched to external division. A quick way to decide which situation a question describes is to ask whether the ratio point is described as lying "between" the two given points (internal) or "on the line produced" / "outside the segment" (external) — ISC problems usually signal this in words before you touch any algebra.

Find the coordinates of the point that divides the segment joining $A(2,-3)$ and $B(-4,6)$ internally in the ratio $1:2$.

Step 1. Identify the values. Here $x_1=2,\ y_1=-3,\ x_2=-4,\ y_2=6$, and the ratio is $m:n = 1:2$, so $m=1$, $n=2$.

Step 2. Substitute into the $x$-coordinate formula. $x = \dfrac{mx_2+nx_1}{m+n} = \dfrac{(1)(-4)+(2)(2)}{1+2} = \dfrac{-4+4}{3} = 0$.

Step 3. Substitute into the $y$-coordinate formula. $y = \dfrac{my_2+ny_1}{m+n} = \dfrac{(1)(6)+(2)(-3)}{1+2} = \dfrac{6-6}{3} = 0$.

Step 4. State and sanity-check the answer. The dividing point is $R(0,0)$, the origin. Since $R$ carries ratio $1$ (nearer $B$) to $2$ (nearer $A$) reversed correctly against $m$ near $Q=B$ and $n$ near $P=A$, $R$ should sit closer to $A$ than to $B$ — and $(0,0)$ does lie closer to $A(2,-3)$ than to $B(-4,6)$, consistent with the numbers found. The dividing point is $R(0,0)$.

Competitive stretch

In what ratio does $P(2,5)$ divide the segment joining $A(8,2)$ and $B(-6,9)$, and is the division internal or external?

Step 1. Let the ratio be $k:1$ — using $k:1$ instead of $m:n$ leaves only one unknown to solve for. Apply the internal-division formula to the $x$-coordinate: $2 = \dfrac{k(-6)+1(8)}{k+1}$.

Step 2. Solve for $k$. $2(k+1) = -6k+8 \implies 2k+2 = -6k+8 \implies 8k=6 \implies k = \tfrac34$. Since $k>0$, the division is internal, in the ratio $3:4$.

Step 3. Check with $y$: $y = \dfrac{(3/4)(9)+(1)(2)}{3/4+1} = \dfrac{27/4+2}{7/4} = \dfrac{35/4}{7/4} = 5$, matching the given $y$-coordinate of $P$. The ratio is $3:4$, internal.

• Mixing up $m$ and $n$, or which point is "first." The formula is not symmetric in $P$ and $Q$: swapping them without also swapping $m$ and $n$ gives a different point. Always match $m$ (near $Q$) and $n$ (near $P$) to the order written in the ratio $m:n$.
• Using the internal formula when the point actually lies outside the segment. If the $k:1$ method returns a negative $k$, that signals external division — switch to the minus-sign version of the formula.
• Forgetting the mid-point formula is just $m=n$. Treating mid-point and section formula as unrelated facts leads to mixing up which denominator to use.
• Sign errors with negative coordinates. Substitute carefully into $mx_2+nx_1$ (not $mx_1+nx_2$), and keep negative signs attached to the correct coordinate throughout.
• Assuming a ratio must be given as whole numbers. The $k:1$ method deliberately produces fractional or decimal values of $k$; a ratio such as $3:4$ is exact, not an approximation to round off.

Ask the tutor: If you want to see the section formula applied inside a bigger problem — like finding where an angle bisector meets a triangle's side — ask the AI tutor to walk through one with you.