Section Formula Class 10 with Worked Examples

Key Points

• The section formula class 10 for internal division is $\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$ for a point dividing $AB$ in ratio $m:n$.

• The weights cross over: $m$, the first ratio term, multiplies $x_2$ from the second point, not $x_1$.

• For external division the signs become subtractions, giving $\left(\frac{mx_2-nx_1}{m-n}, \frac{my_2-ny_1}{m-n}\right)$.

• Setting $m = n = 1$ reduces the formula to the midpoint of the segment, so the midpoint is a special case rather than a separate rule.

• The ratio is written from the first named point, so $m:n$ on segment $AB$ means the point is $m$ parts from $A$ and $n$ parts from $B$.

What Is the Section Formula Class 10?

The section formula class 10 locates the point that divides a line segment in a given ratio. For points $A(x_1, y_1)$ and $B(x_2, y_2)$ and a point $P$ dividing $AB$ internally in ratio $m:n$:

$$P = \left(\frac{mx_2 + nx_1}{m+n}, \ \frac{my_2 + ny_1}{m+n}\right)$$

The method belongs to coordinate geometry, the field MacTutor credits to Descartes for his application of algebra to geometry, published in La Géométrie at Leiden in 1637.

Why the Weights Cross Over

The formula is a weighted average, and the weight attached to each point is the ratio part belonging to the other end. That is the single most important structural fact in it.

Think about a point very close to $A$. Then it is a tiny distance from $A$ and almost the whole length from $B$, so the ratio $m:n$ has $m$ small and $n$ large. For the answer to sit near $A$, the large weight $n$ must multiply $x_1$, which is $A$'s coordinate.

So $n$ goes with $x_1$ and $m$ goes with $x_2$. The crossing is not an arbitrary convention, it is what makes a small $m$ pull the point towards $A$ rather than away from it.

Variable Key

Examples of the Section Formula Class 10

Example 1

Find the point dividing the segment from $A(1,2)$ to $B(6,7)$ internally in the ratio 2:3.

$$x = \frac{2(6)+3(1)}{2+3} = \frac{15}{5} = 3, \qquad y = \frac{2(7)+3(2)}{5} = \frac{20}{5} = 4$$

Final answer: $P(3, 4)$

Example 2

Find the same point, pairing $m$ with the first point instead.

Reading the formula as "$m$ goes with the first coordinate" gives

$$x = \frac{2(1)+3(6)}{5} = \frac{20}{5} = 4, \qquad y = \frac{2(2)+3(7)}{5} = \frac{25}{5} = 5$$

That returns $(4,5)$. Check it against the ratio: the point should be only 2 parts out of 5 from $A$, so it must sit nearer to $A$ than to $B$. The point $(4,5)$ sits nearer to $B(6,7)$, so the pairing is reversed.

Crossing the weights correctly gives $(3,4)$, which is two fifths of the way along and comfortably nearer $A$.

Final answer: $P(3,4)$, and a quick "which end should it be nearer" check catches the swap every time.

Example 3

Find the midpoint of $A(-2, 5)$ and $B(4, -1)$ using the section formula.

A midpoint divides in the ratio 1:1, so $m = n = 1$:

$$x = \frac{1(4)+1(-2)}{2} = 1, \qquad y = \frac{1(-1)+1(5)}{2} = 2$$

Final answer: $(1, 2)$, which is the midpoint formula's answer as expected.

Example 4

Find the point dividing $A(2,1)$ to $B(8,4)$ externally in the ratio 3:1.

External division subtracts:

$$x = \frac{3(8)-1(2)}{3-1} = \frac{22}{2} = 11, \qquad y = \frac{3(4)-1(1)}{2} = \frac{11}{2} = 5.5$$

Final answer: $(11, 5.5)$

The point lies beyond $B$, outside the segment, which is what external division means.

Example 5

In what ratio does the point $(4, 5)$ divide the segment from $A(1, 2)$ to $B(7, 8)$?

Let the ratio be $k:1$ and solve using the $x$-coordinate:

$$4 = \frac{k(7)+1(1)}{k+1} \implies 4k+4 = 7k+1 \implies k = 1$$

Final answer: The ratio is $1:1$, so $(4,5)$ is the midpoint. Checking the $y$-coordinate confirms it: $\frac{8+2}{2} = 5$.

Internal, External and the Midpoint Compared

The centroid row is the section formula applied twice. A triangle's centroid divides each median in the ratio 2:1 from the vertex, which is why its coordinates come out as a plain average of all three vertices.

Where Do Students Trip Up on the Section Formula?

Mistake 1: Pairing m with the first point

Where it slips in: The ratio is written $m:n$ and the points are listed $A$ then $B$, so the natural reading lines them up in order.

Don't do this: $\frac{mx_1+nx_2}{m+n}$

The correct way: $\frac{mx_2+nx_1}{m+n}$. The weights cross, so $m$ multiplies $B$'s coordinate.

Students who ask which end the answer should be nearer, before computing, catch the swap immediately. If $m$ is smaller than $n$ the point sits nearer $A$, and any answer that lands nearer $B$ has the pairing reversed.

Mistake 2: Using the internal formula for external division

Where it slips in: Both versions look almost identical, differing only in two signs.

Don't do this: Add in the numerator and denominator when the question says externally.

The correct way: External division subtracts in both, giving $\frac{mx_2-nx_1}{m-n}$. An external point must land outside the segment, so an answer between $A$ and $B$ signals the wrong version.

Mistake 3: Reversing the ratio order

Where it slips in: A ratio of 2:3 on segment $AB$ gets read as 3:2, which is the same division measured from the other end.

Don't do this: Treat $2:3$ from $A$ as interchangeable with $2:3$ from $B$.

The correct way: The first term always belongs to the first named point. Dividing $AB$ in 2:3 is the same point as dividing $BA$ in 3:2, so the segment's direction must be fixed before the numbers go in.

Bottom Line

• The section formula class 10 is $\left(\frac{mx_2+nx_1}{m+n}, \frac{my_2+ny_1}{m+n}\right)$ for internal division in ratio $m:n$.

• The weights cross over, so the first ratio term multiplies the second point's coordinates.

• External division replaces both additions with subtractions and places the point outside the segment.

• The midpoint is the $1:1$ case, not a separate formula.

• To work through coordinate geometry with a teacher, explore a geometry tutor or math classes online.

Where to Go From Here

Find the point dividing $A(-1, 3)$ to $B(5, -3)$ internally in 1:2. Then find the external 3:2 division of the same segment, and confirm it lies outside it.

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Read More

• Mid Point Theorem

• Coordinate Geometry

• Cartesian Coordinate System

• Distance Formula

• Centroid of a Triangle