MathsICSE Class 10 · Section and Mid-point Formula

Section Formula for Internal Division: Points, Ratios and Solved Questions (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

The point P dividing the join of A(x₁, y₁) and B(x₂, y₂) internally in the ratio m : n is P((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)). Each ratio number multiplies the far end. To find an unknown ratio, take it as k : 1 and solve.

P is 2 parts from A and 3 parts from B, so m = 2 multiplies B's coordinates and n = 3 multiplies A's, giving P(3, 4).

Illustration drawn for QuestionBuddy.

Understand Section Formula (Internal Division)

The section formula locates a point on a line segment from the ratio in which it cuts the segment. The CISCE syllabus includes internal division only.

The formula. If $P$ divides $A B$ internally with $A P : P B = m : n$ , where $A (x_{1}, y_{1})$ and $B (x_{2}, y_{2})$ :

$P = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n}) .$

Why the numbers cross over. $P$ is $\frac{m}{m + n}$ of the way from $A$ to $B$ . So $x_{P} = x_{1} + \frac{m}{m + n} (x_{2} - x_{1}) = \frac{m x _{2} + n x _{1}}{m + n}$ . The larger ratio number goes with the end the point is further from, so it multiplies the other end: $m$ (next to $A$ ) multiplies $B$ 's coordinates.

**Finding the ratio: the $k : 1$ method.** If the ratio is unknown, call it $k : 1$ . Then

$x_{P} = \frac{k x _{2} + x _{1}}{k + 1}$ and $y_{P} = \frac{k y _{2} + y _{1}}{k + 1}$ .

Use whichever coordinate of $P$ is known to find $k$ , then the other to find the missing coordinate. A positive $k$ confirms internal division. If $k$ comes out negative, the point is not between $A$ and $B$ ; the 2024 examiners reported answers like $k = - 3$ from a misapplied formula when the correct value was $k = 3$ .

Where an axis cuts a segment. On the x-axis $y = 0$ ; on the y-axis $x = 0$ . Put that coordinate into the $k : 1$ form.

Working backwards. If $P$ , $A$ and the ratio are known, the section formula gives equations for $B$ 's coordinates.

Fractions of a segment. 'P is $\frac{2}{5}$ of the way from A to B' means $A P : P B = 2 : 3$ . 'P is three times as far from A as from B' means $A P : P B = 3 : 1$ .

The mid-point formula is the special case $m = n$ , and the centroid of a triangle divides each median in $2 : 1$ ; both have their own topic pages.

Section Formula (Internal Division): formulas to remember

Section formula (internal): $P = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$; $A P : P B = m : n$ .
k : 1 form: $x = \frac{k x _{2} + x _{1}}{k + 1}, y = \frac{k y _{2} + y _{1}}{k + 1}$; For an unknown ratio.
Fraction along the segment: $P = A + \frac{m}{m + n} (B - A)$; $P$ is $\frac{m}{m + n}$ of the way from $A$ .
Axis crossing: $x-axis: y = 0; y-axis: x = 0$; Substitute into the $k : 1$ form.

How to solve Section Formula (Internal Division) questions

Label the ends, match each ratio number to its end, then substitute.

1Write $A (x_{1}, y_{1})$ , $B (x_{2}, y_{2})$ and $m : n$ with $m$ next to $A$ .
2Cross-multiply: $m$ with $B$ 's coordinates and $n$ with $A$ 's; divide by $m + n$ .
3For an unknown ratio, write $k : 1$ and use a known coordinate of $P$ to find $k$ .
4Convert $k$ to a whole-number ratio (for example $k = \frac{3}{2}$ gives $3 : 2$ ).
5Find any missing coordinate of $P$ with the ratio.
6Check: $P$ should lie between $A$ and $B$ , nearer the end with the smaller ratio number.

Section Formula (Internal Division) in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Dividing a route in a given ratio

Suppose two friends live at $A (2, 1)$ and $B (11, 7)$ on a town grid (in km) and agree to meet on the straight road between them at a point twice as far from A as from B.

Find the meeting point.

Try it first, then show the solutionSolution

Twice as far from $A$ as from $B$ : $A P : P B = 2 : 1$ .
$P = (\frac{2 \times 11 + 1 \times 2}{3}, \frac{2 \times 7 + 1 \times 1}{3}) = (8, 5)$ .

Answer: At $(8, 5)$ .

Memory hook: Twice as far from A means the ratio 2 sits next to A.

A point a fraction of the way along

Imagine a rope stretched from the top of a pole at $A (0, 12)$ to a peg at $B (16, 0)$ (in metres), with a lamp hung one quarter of the way from A.

Find the lamp's position.

Try it first, then show the solutionSolution

One quarter of the way from $A$ : $A P : P B = 1 : 3$ .
$P = (\frac{1 \times 16 + 3 \times 0}{4}, \frac{1 \times 0 + 3 \times 12}{4}) = (4, 9)$ .

Answer: At $(4, 9)$ : 4 m across and 9 m high.

Memory hook: A fraction $\frac{m}{m + n}$ of the way means the ratio $m : n$ .

Tricks to remember Section Formula (Internal Division)

Short memory aids. Say each one aloud once, then use it on a question the same day.

Criss-cross
Write $m : n$ above $A$ and $B$ , then multiply diagonally: $m$ times $B$ , $n$ times $A$ .
Use it when: Getting the numerators right.
k : 1 for the unknown
One unknown ratio, one letter: $k$ .
Use it when: Finding a ratio from one known coordinate.
Axis means zero
Cuts the x-axis: $y = 0$ . Cuts the y-axis: $x = 0$ .
Use it when: Ratio in which an axis divides a segment.

Section Formula (Internal Division): solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Dividing in 1 : 2 · 1 mark · MCQBoard level

The point dividing the join of $(1, 2)$ and $(7, 11)$ internally in the ratio $1 : 2$ is:
(a) $(3, 5)$
(b) $(5, 8)$
(c) $(4, 6.5)$
(d) $(3, 4)$

Try it first, then show the solutionSolution

$x = \frac{1 \times 7 + 2 \times 1}{3} = 3$ , $y = \frac{1 \times 11 + 2 \times 2}{3} = 5$ .
Option (b) uses the ratio the wrong way round; (c) is the mid-point.

Answer: (a) $(3, 5)$

Example 2: Dividing in 3 : 2 · 3 marks · Short answerBoard level

Find the coordinates of the point that divides the join of $A (- 2, 5)$ and $B (8, - 5)$ internally in the ratio $3 : 2$ .

Try it first, then show the solutionSolution

$x = \frac{3 \times 8 + 2 \times ( - 2 )}{5} = \frac{24 - 4}{5} = 4$ .
$y = \frac{3 \times ( - 5 ) + 2 \times 5}{5} = \frac{- 15 + 10}{5} = - 1$ .

Answer: $(4, - 1)$

Example 3: Finding the ratio and a coordinate · 3 marks · Short answerHarder board level

In what ratio does the point $P (2, y)$ divide the join of $A (- 4, 1)$ and $B (6, - 4)$ ? Find $y$ .

Try it first, then show the solutionSolution

Let the ratio be $k : 1$ . Then $2 = \frac{6 k - 4}{k + 1} \Rightarrow 2 k + 2 = 6 k - 4 \Rightarrow k = \frac{3}{2}$ . Ratio $3 : 2$ .
$y = \frac{3 \times ( - 4 ) + 2 \times 1}{5} = \frac{- 10}{5} = - 2$ .

Answer: $3 : 2$ ; $y = - 2$

Example 4: Where the y-axis cuts a segment · 4 marks · Short answerHarder board level

Find the ratio in which the y-axis divides the join of $A (- 3, - 4)$ and $B (6, 2)$ , and the point where it cuts.

Try it first, then show the solutionSolution

On the y-axis $x = 0$ : $\frac{6 k - 3}{k + 1} = 0 \Rightarrow k = \frac{1}{2}$ , so the ratio is $1 : 2$ .
$y = \frac{1 \times 2 + 2 \times ( - 4 )}{3} = \frac{- 6}{3} = - 2$ .
The point is $(0, - 2)$ .

Answer: $1 : 2$ ; at $(0, - 2)$

Example 5: Finding an end point · 3 marks · Short answerHOTS / competency

The point $P (1, - 2)$ divides the join of $A (- 3, 4)$ and $B$ internally in the ratio $2 : 3$ . Find $B$ .

Try it first, then show the solutionSolution

Let $B = (p, q)$ . $1 = \frac{2 p + 3 ( - 3 )}{5} \Rightarrow 2 p - 9 = 5 \Rightarrow p = 7$ .
$- 2 = \frac{2 q + 3 \times 4}{5} \Rightarrow 2 q + 12 = - 10 \Rightarrow q = - 11$ .

Answer: $B (7, - 11)$

Example 6: Case study: a zip-line · 4 marks · Case-basedHOTS / competency

On a side view drawn in metres, a zip-line runs straight from the top of a tower at $A (0, 30)$ to a landing point $B (60, 0)$ .
(a) A camera is fixed one third of the way from $A$ . Find its position.
(b) A rest stop divides $A B$ in the ratio $3 : 1$ . Find it.
(c) The cable passes directly above a tree at $x = 48$ . In what ratio does that point divide $A B$ , and how high is the cable there?

Try it first, then show the solutionSolution

(a) Ratio $1 : 2$ : $(\frac{60}{3}, \frac{0 + 60}{3}) = (20, 20)$ .
(b) Ratio $3 : 1$ : $(\frac{180}{4}, \frac{0 + 30}{4}) = (45, 7.5)$ .
(c) $\frac{60 k}{k + 1} = 48 \Rightarrow 12 k = 48 \Rightarrow k = 4$ : ratio $4 : 1$ . Height $= \frac{4 \times 0 + 1 \times 30}{5} = 6$ .

Answer: (a) $(20, 20)$ (b) $(45, 7.5)$ (c) $4 : 1$ ; 6 m high

Where students lose marks in Section Formula (Internal Division)

The slips examiners see most often, and what to write instead.

Mistake: Multiplying each ratio number by its own end
$m$ is next to $A$ , but multiplies $B$ 's coordinates.
Wrong working
$1 : 2$ from $(1, 2)$ to $(7, 11)$ : $(\frac{1 + 14}{3}, \frac{2 + 22}{3}) = (5, 8)$
Right working
$(\frac{7 + 2}{3}, \frac{11 + 4}{3}) = (3, 5)$
Do this: Criss-cross, and check the point is nearer the end with the smaller ratio number.
Usually costs: All accuracy marks.
Mistake: A negative k from a misapplied formula
For internal division $k$ must be positive.
Wrong working
$\frac{- 4 k + 0}{k + 1} = - 3$ solved as $k = - 3$ .
Right working
$- 4 k = - 3 k - 3 \Rightarrow k = 3$ .
Do this: If $k$ is negative, recheck the algebra.
Usually costs: Reported in 2024 Q7(iii)(c).
Mistake: Swapping x and y
A point on the y-axis has $x = 0$ .
Wrong working
$A$ on the y-axis written as $(6, 0)$ .
Right working
$A (0, 6)$ .
Do this: Say 'on the y-axis, x is zero' as you write.
Usually costs: Reported in 2024 Q7(iii)(a); every later part uses the wrong point.

Exam tips

2024 Q7(iii) (4 marks) read $A$ and $D$ off the axes, then found $k$ from $\frac{- 4 k + 0}{k + 1} = - 3$ , giving $k = 3$ .
State the ratio in its simplest whole-number form, such as $3 : 2$ , not $1.5 : 1$ .
Show the formula with letters first; the method mark is for correct substitution.
External division is not in the ICSE syllabus; every answer should lie between the two ends.

Practice questions on Section Formula (Internal Division)

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
The point dividing the join of $(0, 0)$ and $(10, - 5)$ internally in the ratio $2 : 3$ is:
(a) $(6, - 3)$
(b) $(4, - 2)$
(c) $(5, - 2.5)$
(d) $(2, - 1)$
Hint
$\frac{2}{5}$ of the way from the origin.
Answer
(b) $(4, - 2)$
Q2 · 2 marks · Short answerBoard level
Find the point dividing the join of $(- 1, 7)$ and $(4, - 3)$ internally in the ratio $2 : 3$ .
Hint
$x = \frac{8 - 3}{5}$ .
Answer
$(1, 3)$
Q3 · 3 marks · Short answerHarder board level
Find the ratio in which the x-axis divides the join of $A (5, - 6)$ and $B (- 1, 3)$ , and the point of division.
Hint
Set $\frac{3 k - 6}{k + 1} = 0$ .
Answer
$k = 2$ , ratio $2 : 1$ ; point $(\frac{- 2 + 5}{3}, 0) = (1, 0)$ .
Q4 · 3 marks · Short answerHarder board level
$P$ lies on the segment from $A (- 3, 2)$ to $B (7, - 8)$ with $A P = \frac{3}{5} A B$ . Find $P$ .
Hint
$A P : P B = 3 : 2$ .
Answer
$(3, - 4)$
Q5 · 1 mark · Assertion-reasonHarder board level
Assertion (A): The point $(2, 3)$ divides the join of $(1, 1)$ and $(4, 7)$ in the ratio $1 : 2$ .
Reason (R): The point dividing the join of $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$ in the ratio $m : n$ is $(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$ .
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
$(\frac{4 + 2}{3}, \frac{7 + 2}{3})$ .
Answer
(c)
Q6 · 4 marks · Short answerHOTS / competency
In what ratio does the line $x - y - 2 = 0$ divide the join of $(3, - 1)$ and $(8, 9)$ ? Find the point of division.
Hint
The point $(\frac{8 k + 3}{k + 1}, \frac{9 k - 1}{k + 1})$ satisfies $x - y = 2$ .
Answer
$8 k + 3 - 9 k + 1 = 2 k + 2$ gives $k = \frac{2}{3}$ : ratio $2 : 3$ ; point $(5, 3)$ .
Q7 · 3 marks · Short answerHarder board level
$A (1, 3)$ and $B (9, - 5)$ are given. Find the point $P$ on $A B$ that is three times as far from $A$ as from $B$ .
Hint
$A P : P B = 3 : 1$ .
Answer
$P = (\frac{27 + 1}{4}, \frac{- 15 + 3}{4}) = (7, - 3)$

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Write the section formula for internal division.Show answer
$(\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})$
2.Point dividing $(0, 0)$ and $(6, 9)$ in $2 : 1$ ?Show answer
$(4, 6)$
3.What ratio does 'one third of the way from A' mean?Show answer
$A P : P B = 1 : 2$
4.Which coordinate is 0 where a segment meets the x-axis?Show answer
The y-coordinate.
5.If $k = \frac{5}{3}$ , what is the ratio?Show answer
$5 : 3$

Section Formula (Internal Division): frequently asked questions

What is the section formula?

The point dividing the join of (x₁, y₁) and (x₂, y₂) internally in the ratio m : n is ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n)).

How do I find the ratio in which a point divides a segment?

Take the ratio as k : 1, write one coordinate of the point using the section formula, set it equal to the known value, and solve for k.

Is external division in the ICSE Class 10 syllabus?

No. The CISCE syllabus says internal section only.