Section formula

Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ be two fixed points, and let $R(x,y)$ lie on segment $PQ$, dividing it internally so that $PR:RQ = m_1:m_2$, measured starting from $P$ and moving toward $Q$. The section formula gives $R$'s coordinates directly, without drawing anything:

where: $(x_1,y_1)$ and $(x_2,y_2)$ are the given endpoints, $m_1:m_2$ is the ratio in which $R$ divides $PQ$ (from $P$ toward $Q$), and $(x,y)$ are the coordinates of $R$.

Why $x$ and $y$ split apart cleanly. Drop perpendiculars from $P$, $R$, $Q$ down to the $x$-axis. Because all three perpendiculars are parallel, the two small triangles formed (one on each side of $R$) are similar by the AA criterion — matching angles come from the parallel drops and the shared line $PQ$. That similarity is what lets the ratio $m_1:m_2$ act on the $x$-coordinates and the $y$-coordinates completely independently: you can compute $x$ first and $y$ second in either order, and an arithmetic slip in one never touches the other.

Midpoint as a special case. When $R$ is exactly halfway, $PR:RQ = 1:1$, so $m_1=m_2$. Substituting $m_1=m_2=1$ collapses the section formula to

the plain average of the two $x$-coordinates and the two $y$-coordinates. It is worth memorising separately, since it appears constantly — finding the midpoint of a diagonal, or using the fact that a parallelogram's diagonals bisect each other to locate a missing vertex. A quick sanity check before trusting any answer: if $m_1$ is much larger than $m_2$, $R$ should sit close to $Q$ (the far point); if $m_1$ is much smaller than $m_2$, $R$ should sit close to $P$. Running this mentally after computing $R(x,y)$ catches a swapped ratio before it costs marks. This formula also quietly extends any "find a point" question into a small toolkit. A ratio can be handed to you directly ($m_1:m_2$ stated outright), or disguised as a fraction of the whole segment (e.g. "$R$ is $\tfrac{2}{9}$ of the way from $P$ to $Q$", which converts to a ratio before you touch the formula), or reversed into "given the dividing point and one endpoint, find the other endpoint" — the same equations, just solved for a different unknown. Recognising which of these three shapes a question takes is often the real skill being tested, not the substitution itself.

Find the coordinates of the point which divides the line segment joining $A(4,-1)$ and $B(-2,8)$ internally in the ratio $2:1$, and check the answer using the midpoint idea.

Step 1. Identify the values: $(x_1,y_1)=(4,-1)$, $(x_2,y_2)=(-2,8)$, $m_1=2$, $m_2=1$.

Step 2. Apply the section formula for $x$: $x = \dfrac{m_1x_2+m_2x_1}{m_1+m_2} = \dfrac{2(-2)+1(4)}{2+1} = \dfrac{-4+4}{3} = 0$.

Step 3. Apply the section formula for $y$: $y = \dfrac{m_1y_2+m_2y_1}{m_1+m_2} = \dfrac{2(8)+1(-1)}{2+1} = \dfrac{16-1}{3} = 5$. So the required point is $(0,5)$.

Step 4. Sanity-check with the reverse question: if $C(1,-2)$ has midpoint $M(3,4)$ with an unknown second point $D(x,y)$, the midpoint formula gives $3=\dfrac{1+x}{2}$ so $x=5$, and $4=\dfrac{-2+y}{2}$ so $y=10$; hence $D=(5,10)$. Checking, the midpoint of $C(1,-2)$ and $D(5,10)$ is $\left(\dfrac{1+5}{2},\dfrac{-2+10}{2}\right)=(3,4)$, which matches $M$. Both results confirm the same formula working forward, $(0,5)$, and backward, $D=(5,10)$.

Competitive stretch

A point $R$ divides the segment joining $P(-3, 10)$ and $Q(6, -8)$ such that $PR = \frac{2}{9}PQ$. Find $R$.

Step 1. Convert the fractional statement into a ratio. If $PR = \frac{2}{9}PQ$, then $RQ = PQ - PR = \frac{7}{9}PQ$, so $PR:RQ = \frac{2}{9}:\frac{7}{9} = 2:7$. This means $m_1=2$, $m_2=7$.

Step 2. Apply the section formula for $x$: $x = \dfrac{2(6)+7(-3)}{2+7} = \dfrac{12-21}{9} = \dfrac{-9}{9} = -1$.

Step 3. Apply the section formula for $y$: $y = \dfrac{2(-8)+7(10)}{2+7} = \dfrac{-16+70}{9} = \dfrac{54}{9} = 6$. So $R = (-1, 6)$. The extra step here is entirely in reading "$PR = \frac{2}{9}PQ$" correctly into a ratio $m_1:m_2$ before ever touching the formula — the formula itself works exactly as before.

• Mixing up the order of the ratio. For ratio $m_1:m_2$, $m_1$ multiplies $x_2$ (the second point) and $m_2$ multiplies $x_1$ (the first point). Swapping them gives a different, wrong point.
• Forgetting which point is $P$ and which is $Q$. The ratio $m_1:m_2$ means $PR:RQ$ starting from the first named point; relabelling the two given points changes the answer.
• Treating the midpoint formula as a separate rule. It is just the section formula with $m_1=m_2=1$ — no need to memorise it independently, though it is worth knowing by heart for speed.
• Sign errors with negative coordinates. Track signs carefully inside the numerator before dividing by $m_1+m_2$; a dropped negative sign is the single most common slip.

Ask the tutor: If you're unsure which point should be $m_1$ and which should be $m_2$, or want another ratio worked through step by step, ask the AI tutor.