Question 1

The mid-point of the segment joining $(2,3)$ and $(4,7)$ is:

Accepted Answer

**Answer: (a) $(3,5)$.** $\left(\dfrac{2+4}{2},\dfrac{3+7}{2}\right)=(3,5).$

Question 2

The centroid of the triangle with vertices $(0,0)$, $(6,0)$ and $(0,9)$ is:

Accepted Answer

**Answer: (a) $(2,3)$.** $\left(\dfrac{0+6+0}{3},\dfrac{0+0+9}{3}\right)=(2,3).$

Question 3

The point dividing the join of $(1,2)$ and $(7,5)$ internally in the ratio $1:2$ is:

Accepted Answer

**Answer: (a) $(3,3)$.** $\left(\dfrac{1\cdot7+2\cdot1}{3},\dfrac{1\cdot5+2\cdot2}{3}\right)=\left(\dfrac{9}{3},\dfrac{9}{3}\right)=(3,3).$

Question 4

The ratio in which the $x$-axis divides the join of $(2,-3)$ and $(5,6)$ is:

Accepted Answer

**Answer: (a) $1:2$.** Let the ratio be $k:1$. On the $x$-axis $y=0$: $\dfrac{6k+(-3)}{k+1}=0\Rightarrow 6k=3\Rightarrow k=	frac12$, i.e. $1:2$.

Question 5

**Assertion (A):** The mid-point of the segment joining $(-1,4)$ and $(3,-2)$ is $(1,1)$. **Reason (R):** The mid-point of $(x_1,y_1)$ and $(x_2,y_2)$ is $\left(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\right)$.

Accepted Answer

**Answer: (a)** $\left(\dfrac{-1+3}{2},\dfrac{4+(-2)}{2}\right)=(1,1)$, so A is true and R is the formula that produces it.

Question 6

Find the mid-point of the segment joining $A(-3,5)$ and $B(7,-1)$.

Accepted Answer

$M=\left(\dfrac{-3+7}{2},\dfrac{5+(-1)}{2}\right)=\left(\dfrac{4}{2},\dfrac{4}{2}\right)=(2,2).$

Question 7

Find the coordinates of the point which divides the join of $(-1,3)$ and $(4,-7)$ internally in the ratio $2:3$.

Accepted Answer

$\left(\dfrac{2\cdot4+3\cdot(-1)}{2+3},\dfrac{2\cdot(-7)+3\cdot3}{2+3}\right)=\left(\dfrac{8-3}{5},\dfrac{-14+9}{5}\right)=\left(\dfrac{5}{5},\dfrac{-5}{5}\right)=(1,-1).$

Question 8

The mid-point of the segment joining $(2a,4)$ and $(-2,3b)$ is $(1,5)$. Find the values of $a$ and $b$.

Accepted Answer

$x$-coordinate: $\dfrac{2a+(-2)}{2}=1\Rightarrow 2a-2=2\Rightarrow a=2.$ $y$-coordinate: $\dfrac{4+3b}{2}=5\Rightarrow 4+3b=10\Rightarrow 3b=6\Rightarrow b=2.$

Question 9

Find the ratio in which the point $(-4,6)$ divides the join of $A(-6,10)$ and $B(3,-8)$.

Accepted Answer

Let the ratio be $k:1$. Using the $x$-coordinate: $\dfrac{3k+(-6)}{k+1}=-4\Rightarrow 3k-6=-4k-4\Rightarrow 7k=2\Rightarrow k=\dfrac{2}{7}.$ So the ratio is $2:7$. (Check with $y$: $\dfrac{-8(2)+10(7)}{2+7}=\dfrac{-16+70}{9}=6$ correct.)

Question 10

Two vertices of a triangle are $(1,2)$ and $(3,5)$ and its centroid is $(3,3)$. Find the third vertex.

Accepted Answer

Let the third vertex be $(x,y)$. $\dfrac{1+3+x}{3}=3\Rightarrow 4+x=9\Rightarrow x=5.$ $\dfrac{2+5+y}{3}=3\Rightarrow 7+y=9\Rightarrow y=2.$ The third vertex is $(5,2).$

Question 11

Points $A(-3,4)$ and $B(9,-2)$ are given. Find (i) the mid-point $M$ of $AB$, (ii) the point $P$ that divides $AB$ in the ratio $1:2$, and (iii) the point $Q$ that divides $AB$ in the ratio $2:1$.

Accepted Answer

(i) $M=\left(\dfrac{-3+9}{2},\dfrac{4+(-2)}{2}\right)=(3,1).$

(ii) $P=\left(\dfrac{1\cdot9+2\cdot(-3)}{3},\dfrac{1\cdot(-2)+2\cdot4}{3}\right)=\left(\dfrac{3}{3},\dfrac{6}{3}\right)=(1,2).$

(iii) $Q=\left(\dfrac{2\cdot9+1\cdot(-3)}{3},\dfrac{2\cdot(-2)+1\cdot4}{3}\right)=\left(\dfrac{15}{3},\dfrac{0}{3}\right)=(5,0).$

Question 12

Three vertices of a parallelogram $ABCD$ taken in order are $A(1,2)$, $B(4,3)$ and $C(6,6)$. (i) Using the fact that the diagonals of a parallelogram bisect each other, find the coordinates of $D$. (ii) Find the coordinates of the point where the diagonals intersect.

Accepted Answer

(i) In parallelogram $ABCD$ the diagonals $AC$ and $BD$ bisect each other, so their mid-points coincide.

Mid-point of $AC=\left(\dfrac{1+6}{2},\dfrac{2+6}{2}\right)=\left(\dfrac{7}{2},4\right).$

Let $D=(x,y)$ . Mid-point of $BD=\left(\dfrac{4+x}{2},\dfrac{3+y}{2}\right).$

Equating: $\dfrac{4+x}{2}=\dfrac{7}{2}\Rightarrow x=3$ and $\dfrac{3+y}{2}=4\Rightarrow y=5.$ So $D=(3,5).$

(ii) The diagonals meet at their common mid-point $\left(\dfrac{7}{2},4\right)$ , i.e. $(3.5,\,4).$

Question 13

On a town map drawn with coordinate axes, the library is at $A(2,1)$ and the stadium is at $B(10,7)$ .

(i) A bus stop is built at the mid-point of $AB$ . Find its coordinates.

(ii) A shop stands at the point dividing $A$ to $B$ in the ratio $1:3$ . Find its coordinates.

(iii) A cafe stands at the point dividing $A$ to $B$ in the ratio $3:1$ . Find its coordinates.

Accepted Answer

(i) Bus stop $=\left(\dfrac{2+10}{2},\dfrac{1+7}{2}\right)=(6,4).$

(ii) Shop $=\left(\dfrac{1\cdot10+3\cdot2}{4},\dfrac{1\cdot7+3\cdot1}{4}\right)=\left(\dfrac{16}{4},\dfrac{10}{4}\right)=(4,\,2.5).$

(iii) Cafe $=\left(\dfrac{3\cdot10+1\cdot2}{4},\dfrac{3\cdot7+1\cdot1}{4}\right)=\left(\dfrac{32}{4},\dfrac{22}{4}\right)=(8,\,5.5).$

Question 14

Are these Section and Mid-Point Formula important questions free?

Accepted Answer

Yes. All 13 ICSE Class 10 Maths important questions for Section and Mid-Point Formula are free, with full model answers and no login required.

Question 15

Do these Section and Mid-Point Formula questions follow the latest ICSE syllabus?

Accepted Answer

Yes — they are aligned to the CISCE latest syllabus syllabus for ICSE Class 10 Maths, so nothing here is outside the current course.

Question 16

How should I practise the Section and Mid-Point Formula important questions?

Accepted Answer

Attempt each question on paper first, then reveal the model answer to check your method — not just the final result. Re-do anything you got wrong the same day.

Question 17

What types of questions are covered for Section and Mid-Point Formula?

Accepted Answer

A full mix — multiple-choice questions, assertion–reason questions, very short answer questions, short answer questions, long answer questions, case-based questions — so every format in the ICSE paper is covered.

Question type	Count	Marks
MCQ	4	1
Assertion–Reason	1	1
Very Short	2	2
Short Answer	3	3
Long Answer	2	5
Case-based	1	4

Section and Mid-Point Formula — Important Questions

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