# SELINA Solutions for Class 10 Maths Chapter 13 - Section and Mid-Point Formula

Prepare for a high score in your exam with Selina Solutions for ICSE Class 10 Mathematics Chapter 13 Section and Mid-Point Formula. Get thorough knowledge of concepts to find the co-ordinates of a given point of intersection or trisection. Understand how experts use a graph to find the area of a quadrilateral according to the given data.

With TopperLearning’s Selina solutions, practise applying the section formula and the mid-point formula to solve Maths problems from this chapter. If you wish to score more than average marks, you need to put in more efforts in exam preparation. Our video lessons, Frank solutions, online practice tests and other resources can help you push your efforts in reaching your target score.

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## Chapter 13 - Section and Mid-Point Formula Exercise Ex. 13(A)

Question 1

Calculate the co-ordinates of the point P which divides the line segment joining:

(i) A (1, 3) and B (5, 9) in the ratio 1: 2.

(ii) A (-4, 6) and B (3, -5) in the ratio 3: 2.

Solution 1

(i) Let the co-ordinates of the point P be (x, y).

Thus, the co-ordinates of point P are.

(ii) Let the co-ordinates of the point P be (x, y).

Thus, the co-ordinates of point P are.

Question 2

In what ratio is the line joining (2, -3) and (5, 6) divided by the x-axis.

Solution 2

Let the line joining points A (2, -3) and B (5, 6) be divided by point P (x, 0) in the ratio k: 1.

Thus, the required ratio is 1: 2.

Question 3

In what ratio is the line joining (2, -4) and (-3, 6) divided by the y-axis.

Solution 3

Let the line joining points A (2, -4) and B (-3, 6) be divided by point P (0, y) in the ratio k: 1.

Thus, the required ratio is 2: 3.

Question 4

In what ratio does the point (1, a) divided the join of (-1, 4) and (4, -1)? Also, find the value of a.

Solution 4

Let the point P (1, a) divides the line segment AB in the ratio k: 1.

Using section formula, we have:

Question 5

In what ratio does the point (a, 6) divide the join of (-4, 3) and (2, 8)? Also, find the value of a.

Solution 5

Let the point P (a, 6) divides the line segment joining A (-4, 3) and B (2, 8) in the ratio k: 1.

Using section formula, we have:

Question 6

In what ratio is the join of (4, 3) and (2, -6) divided by the x-axis. Also, find the co-ordinates of the point of intersection.

Solution 6

Let the point P (x, 0) on x-axis divides the line segment joining A (4, 3) and B (2, -6) in the ratio k: 1.

Using section formula, we have:

Thus, the required ratio is 1: 2.

Also, we have:

Thus, the required co-ordinates of the point of intersection are .

Question 7

Find the ratio in which the join of (-4, 7) and (3, 0) is divided by the y-axis. Also, find the coordinates of the point of intersection.

Solution 7

Let S (0, y) be the point on y-axis which divides the line segment PQ in the ratio k: 1.

Using section formula, we have:

Question 8

Points A, B, C and D divide the line segment joining the point (5, -10) and the origin in five equal parts. Find the co-ordinates of A, B, C and D.

Solution 8

Point A divides PO in the ratio 1: 4.

Co-ordinates of point A are:

Point B divides PO in the ratio 2: 3.

Co-ordinates of point B are:

Point C divides PO in the ratio 3: 2.

Co-ordinates of point C are:

Point D divides PO in the ratio 4: 1.

Co-ordinates of point D are:

Question 9

The line joining the points A (-3, -10) and B (-2, 6) is divided by the point P such that Find the co-ordinates of P.

Solution 9

Let the co-ordinates of point P are (x, y).

Question 10

P is a point on the line joining A (4, 3) and B (-2, 6) such that 5AP = 2BP. Find the co-ordinates of P.

Solution 10

5AP = 2BP

The co-ordinates of the point P are

Question 11

Calculate the ratio in which the line joining the points (-3, -1) and (5, 7) is divided by the line x = 2. Also, find the co-ordinates of the point of intersection.

Solution 11

The co-ordinates of every point on the line x = 2 will be of the type (2, y).

Using section formula, we have:

Thus, the required ratio is 5: 3.

Thus, the required co-ordinates of the point of intersection are (2, 4).

Question 12

Calculate the ratio in which the line joining A (6, 5) and B (4, -3) is divided by the line y = 2.

Solution 12

The co-ordinates of every point on the line y = 2 will be of the type (x, 2).

Using section formula, we have:

Thus, the required ratio is 3: 5.

Question 13

The point P (5, -4) divides the line segment AB, as shown in the figure, in the ratio 2: 5. Find the co-ordinates of points A and B.

Solution 13

Point A lies on x-axis. So, let the co-ordinates of A be (x, 0).

Point B lies on y-axis. So, let the co-ordinates of B be (0, y).

P divides AB in the ratio 2: 5.

We have:

Thus, the co-ordinates of point A are (7, 0).

Thus, the co-ordinates of point B are (0, -14).

Question 14

Find the co-ordinates of the points of trisection of the line joining the points (-3, 0) and (6, 6).

Solution 14

Let P and Q be the point of trisection of the line segment joining the points A (-3, 0) and B (6, 6).

So, AP = PQ = QB

We have AP: PB = 1: 2

Co-ordinates of the point P are

We have AQ: QB = 2: 1

Co-ordinates of the point Q are

Question 15

Show that the line segment joining the points (-5, 8) and (10, -4) is trisected by the co-ordinate axes.

Solution 15

Let P and Q be the point of trisection of the line segment joining the points A (-5, 8) and B (10, -4).

So, AP = PQ = QB

We have AP: PB = 1: 2

Co-ordinates of the point P are

We have AQ: QB = 2: 1

Co-ordinates of the point Q are

So, point Q lies on the x-axis.

Hence, the line segment joining the given points A and B is trisected by the co-ordinate axes.

Question 16

Show that A (3, -2) is a point of trisection of the line-segment joining the points (2, 1) and (5, -8). Also, find the co-ordinates of the other point of trisection.

Solution 16

Let A and B be the point of trisection of the line segment joining the points P (2, 1) and Q (5, -8).

So, PA = AB = BQ

We have PA: AQ = 1: 2

Co-ordinates of the point A are

Hence, A (3, -2) is a point of trisection of PQ.

We have PB: BQ = 2: 1

Co-ordinates of the point B are

Question 17

If A = (-4, 3) and B = (8, -6)

(i) Find the length of AB.

(ii) In what ratio is the line joining A and B, divided by the x-axis?

Solution 17

(i) A (-4,3) and B (8, -6)

AB =

(ii) Let P be the point, which divides AB on the x-axis in the ratio k : 1.

Therefore, y-co-ordinate of P = 0.

= 0

-6k + 3 = 0

k =

Required ratio is 1: 2.

Question 18

The line segment joining the points M (5, 7) and N (-3, 2) is intersected by the y-axis at point L. Write down the abscissa of L. Hence, find the ratio in which L divides MN. Also, find the co-ordinates of L.

Solution 18

Since, point L lies on y-axis, its abscissa is 0.

Let the co-ordinates of point L be (0, y). Let L divides MN in the ratio k: 1.

Using section formula, we have:

Thus, the required ratio is 5: 3.

Question 19

A (2, 5), B (-1, 2) and C (5, 8) are the co-ordinates of the vertices of the triangle ABC. Points P and Q lie on AB and AC respectively, such that AP: PB = AQ: QC = 1: 2.

(i) Calculate the co-ordinates of P and Q.

(ii) Show that PQ = BC.

Solution 19

(i) Co-ordinates of P are

Co-ordinates of Q are

(ii) Using distance formula, we have:

BC =

PQ =

Hence, PQ = BC.

Question 20

A (-3, 4), B (3, -1) and C (-2, 4) are the vertices of a triangle ABC. Find the length of line segment AP, where point P lies inside BC, such that BP: PC = 2: 3.

Solution 20

BP: PC = 2: 3

Co-ordinates of P are

Using distance formula, we have:

Question 21

The line segment joining A (2, 3) and B (6, -5) is intercepted by x-axis at the point K. Write down the ordinate of the point K. Hence, find the ratio in which K divides AB. Also, find the co-ordinates of the point K.

Solution 21

Since, point K lies on x-axis, its ordinate is 0.

Let the point K (x, 0) divides AB in the ratio k: 1.

Thus, K divides AB in the ratio 3: 5.

Also, we have:

Thus, the co-ordinates of the point K are .

Question 22

The line segment joining A (4, 7) and B (-6, -2) is intercepted by the y-axis at the point K. Write down the abscissa of the point K. Hence, find the ratio in which K divides AB. Also, find the co-ordinates of the point K.

Solution 22

Since, point K lies on y-axis, its abscissa is 0.

Let the point K (0, y) divides AB in the ratio k: 1.

Thus, K divides AB in the ratio 2: 3.

Also, we have:

Thus, the co-ordinates of the point K are .

Question 23

The line joining P (-4, 5) and Q (3, 2) intersects the y-axis at point R. PM and QN are perpendiculars from P and Q on the x-axis. Find:

(i) the ratio PR: RQ.

(ii) the co-ordinates of R.

(iii) the area of the quadrilateral PMNQ.

Solution 23

(i) Let point R (0, y) divides PQ in the ratio k: 1.

We have:

Thus, PR: RQ = 4: 3

(ii) Also, we have:

Thus, the co-ordinates of point R are .

= (PM + QN) MN

= (5 + 2) 7

= 7 7

= 24.5 sq units

Question 24

In the given figure, line APB meets the x-axis at point A and y-axis at point B. P is the point (-4, 2) and AP: PB = 1: 2. Find the co-ordinates of A and B.

Solution 24

Given, A lies on x-axis and B lies on y-axis.

Let the co-ordinates of A and B be (x, 0) and (0, y) respectively.

Given, P is the point (-4, 2) and AP: PB = 1: 2.

Using section formula, we have:

Thus, the co-ordinates of points A and B are (-6, 0) and (0, 6) respectively.

Question 25

Given a line segment AB joining the points A(-4, 6) and B(8, -3). Find:

(i) the ratio in which AB is divided by the y-axis

(ii) find the coordinates of the point of intersection

(iii) the length of AB

Solution 25

(i)

(ii)

(iii)

Question 26

If P(-b, 9a - 2) divides the line segment joining the points A(-3, 3a + 1) and B(5, 8a) in the ratio 3: 1, find the values of a and b.

Solution 26

Take (x1 , y1) = (-3, 3a + 1) ; (x2 , y2) = B(5, 8a) and

(x, y) = (-b, 9a - 2)

Here m1 = 3 and m2 =1

## Chapter 13 - Section and Mid-Point Formula Exercise Ex. 13(B)

Question 1

Find the mid-point of the line segment joining the points:

(i) (-6, 7) and (3, 5)

(ii) (5, -3) and (-1, 7)

Solution 1

(i) A (-6, 7) and B (3, 5)

Mid-point of AB =

(ii) A (5, -3) and B (-1, 7)

Mid-point of AB =

Question 2

Points A and B have co-ordinates (3, 5) and (x, y) respectively. The mid-point of AB is (2, 3). Find the values of x and y.

Solution 2

Mid-point of AB = (2, 3)

Question 3

A (5, 3), B (-1, 1) and C (7, -3) are the vertices of triangle ABC. If L is the mid-point of AB and M is the mid-point of AC, show that LM = BC.

Solution 3

Given, L is the mid-point of AB and M is the mid-point of AC.

Co-ordinates of L are

Co-ordinates of M are

Using distance formula, we have:

Question 4

Given M is the mid-point of AB, find the co-ordinates of:

(i) A; if M = (1, 7) and B = (-5, 10)

(ii) B; if A = (3, -1) and M = (-1, 3).

Solution 4

(i) Let the co-ordinates of A be (x, y).

Hence, the co-ordinates of A are (7, 4).

(ii) Let the co-ordinates of B be (x, y).

Hence, the co-ordinates of B are (-5, 7).

Question 5

P (-3, 2) is the mid-point of line segment AB as shown in the given figure. Find the co-ordinates of points A and B.

Solution 5

Point A lies on y-axis, so let its co-ordinates be (0, y).

Point B lies on x-axis, so let its co-ordinates be (x, 0).

P (-3, 2) is the mid-point of line segment AB.

Thus, the co-ordinates of points A and B are (0, 4) and (-6, 0) respectively.

Question 6

In the given figure, P (4, 2) is mid-point of line segment AB. Find the co-ordinates of A and B.

Solution 6

Point A lies on x-axis, so let its co-ordinates be (x, 0).

Point B lies on y-axis, so let its co-ordinates be (0, y).

P (4, 2) is mid-point of line segment AB.

Hence, the co-ordinates of points A and B are (8, 0) and (0, 4) respectively.

Question 7

(-5, 2), (3, -6) and (7, 4) are the vertices of a triangle. Find the lengths of its median through the vertex (3, -6)

Solution 7

Let A (-5, 2), B (3, -6) and C (7, 4) be the vertices of the given triangle.

Let AD be the median through A, BE be the median through B and CF be the median through C.

We know that median of a triangle bisects the opposite side.

Co-ordinates of point F are

Co-ordinates of point D are

Co-ordinates of point E are

The median of the triangle through the vertex B(3, -6) is BE

Using distance formula,

Question 8

Given a line ABCD in which AB = BC = CD, B = (0, 3) and C = (1, 8). Find the co-ordinates of A and D.

Solution 8

Given, AB = BC = CD

So, B is the mid-point of AC. Let the co-ordinates of point A be (x, y).

Thus, the co-ordinates of point A are (-1, -2).

Also, C is the mid-point of BD. Let the co-ordinates of point D be (p, q).

Thus, the co-ordinates of point D are (2, 13).

Question 9

One end of the diameter of a circle is (-2, 5). Find the co-ordinates of the other end of it, if the centre of the circle is (2, -1).

Solution 9

We know that the centre is the mid-point of diameter.

Let the required co-ordinates of the other end of mid-point be (x, y).

Thus, the required co-ordinates are (6, -7).

Question 10

A (2, 5), B (1, 0), C (-4, 3) and D (-3, 8) are the vertices of a quadrilateral ABCD. Find the co-ordinates of the mid-points of AC and BD.

Give a special name to the quadrilateral.

Solution 10

Co-ordinates of the mid-point of AC are

Co-ordinates of the mid-point of BD are

Since, mid-point of AC = mid-point of BD

Hence, ABCD is a parallelogram.

Question 11

P (4, 2) and Q (-1, 5) are the vertices of a parallelogram PQRS and (-3, 2) are the co-ordinates of the points of intersection of its diagonals. Find the coordinates of R and S.

Solution 11

Let the coordinates of R and S be (x,y) and (a,b) respectively.

Mid-point of PR is O.

O(-3,2) =

-6 = 4 + x, 4 = 2 + y

x = -10 , y = 2

Hence, R = (-10,2)

Similarly, the mid-point of SQ is O.

Thus, the coordinates of the point R and S are (-10, 2) and (-5, -1).

Question 12

A (-1, 0), B (1, 3) and D (3, 5) are the vertices of a parallelogram ABCD. Find the co-ordinates of vertex C.

Solution 12

Let the co-ordinates of vertex C be (x, y).

ABCD is a parallelogram.

Mid-point of AC = Mid-point of BD

Thus, the co-ordinates of vertex C is (5, 8).

Question 13

The points (2, -1), (-1, 4) and (-2, 2) are mid-points of the sides of a triangle. Find its vertices.

Solution 13

Let A(x1,y1), B and C be the co-ordinates of the vertices of ABC.

Midpoint of AB, i.e. D

Similarly,

Adding (1), (3) and (5), we get,

From (3)

From (5)

Adding (2), (4) and (6), we get,

From (4)

From (6)

Thus, the co-ordinates of the vertices of ABC are (3, 1), (1, -3) and (-5, 7).

Question 14

Points A (-5, x), B (y, 7) and C (1, -3) are collinear (i.e., lie on the same straight line) such that AB = BC. Calculates the values of x and y.

Solution 14

Given, AB = BC, i.e., B is the mid-point of AC.

Question 15

Points P (a, -4), Q (-2, b) and R (0, 2) are collinear. If Q lies between P and R, such that PR = 2QR, calculate the values of a and b.

Solution 15

Given, PR = 2QR

Now, Q lies between P and R, so, PR = PQ + QR

PQ + QR = 2QR

PQ = QR

Q is the mid-point of PR.

Question 16

Calculate the co-ordinates of the centroid of a triangle ABC, if A = (7, -2), B = (0, 1) and C = (-1, 4).

Solution 16

Co-ordinates of the centroid of triangle ABC are

Question 17

The co-ordinates of the centroid of a PQR are (2, -5). If Q = (-6, 5) and R = (11, 8); calculate the co-ordinates of vertex P.

Solution 17

Let G be the centroid of DPQR whose coordinates are (2, -5) and let (x,y) be the coordinates of vertex P.

Coordinates of G are,

6 = x + 5, -15 = y + 13

x = 1, y = -28

Coordinates of vertex P are (1, -28)

Question 18

A (5, x), B (-4, 3) and C (y, -2) are the vertices of the triangle ABC whose centroid is the origin. Calculate the values of x and y.

Solution 18

Given, centroid of triangle ABC is the origin.

## Chapter 13 - Section and Mid-Point Formula Exercise Ex. 13(C)

Question 1

Given a triangle ABC in which A = (4, -4), B = (0, 5) and C = (5, 10). A point P lies on BC such that BP: PC = 3: 2. Find the length of line segment AP.

Solution 1

Given, BP: PC = 3: 2

Using section formula, the co-ordinates of point P are

Using distance formula, we have:

Question 2

A (20, 0) and B (10, -20) are two fixed points. Find the co-ordinates of a point P in AB such that: 3PB = AB. Also, find the co-ordinates of some other point Q in AB such that AB = 6AQ.

Solution 2

Using section formula,

Given, AB = 6AQ

Using section formula,

Question 3

A (-8, 0), B (0, 16) and C (0, 0) are the vertices of a triangle ABC. Point P lies on AB and Q lies on AC such that AP: PB = 3: 5 and AQ: QC = 3: 5. Show that: PQ = BC.

Solution 3

Given that, point P lies on AB such that AP: PB = 3: 5.

The co-ordinates of point P are

Also, given that, point Q lies on AB such that AQ: QC = 3: 5.

The co-ordinates of point Q are

Using distance formula,

Hence, proved.

Question 4

Find the co-ordinates of points of trisection of the line segment joining the point (6, -9) and the origin.

Solution 4

Let P and Q be the points of trisection of the line segment joining A (6, -9) and B (0, 0).

P divides AB in the ratio 1: 2. Therefore, the co-ordinates of point P are

Q divides AB in the ratio 2: 1. Therefore, the co-ordinates of point Q are

Thus, the required points are (4, -6) and (2, -3).

Question 5

A line segment joining A and B (a, 5) is divided in the ratio 1: 3 at P, point where the line segment AB intersects the y-axis.

(i) Calculate the value of 'a'.

(ii) Calculate the co-ordinates of 'P'.

Solution 5

Since, the line segment AB intersects the y-axis at point P, let the co-ordinates of point P be (0, y).

P divides AB in the ratio 1: 3.

Thus, the value of a is 3 and the co-ordinates of point P are.

Question 6

In what ratio is the line joining A (0, 3) and B (4, -1) divided by the x-axis? Write the co-ordinates of the point where AB intersects the x-axis.

Solution 6

Let the line segment AB intersects the x-axis by point P (x, 0) in the ratio k: 1.

Thus, the required ratio in which P divides AB is 3: 1.

Also, we have:

Thus, the co-ordinates of point P are (3, 0).

Question 7

The mid-point of the segment AB, as shown in diagram, is C (4, -3). Write down the co-ordinates of A and B.

Solution 7

Since, point A lies on x-axis, let the co-ordinates of point A be (x, 0).

Since, point B lies on y-axis, let the co-ordinates of point B be (0, y).

Given, mid-point of AB is C (4, -3).

Thus, the co-ordinates of point A are (8, 0) and the co-ordinates of point B are (0, -6).

Question 8

AB is a diameter of a circle with centre C = (-2, 5). If A = (3, -7), find

(i)   the length of radius AC

(ii)  the coordinates of B.

Solution 8

Question 9

Find the co-ordinates of the centroid of a triangle ABC whose vertices are:

A (-1, 3), B (1, -1) and C (5, 1)

Solution 9

Co- ordinates of the centroid of triangle ABC are

Question 10

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

Solution 10

It is given that the mid-point of the line-segment joining (4a, 2b - 3) and (-4, 3b) is (2, -2a).

Question 11

The mid-point of the line segment joining (2a, 4) and (-2, 2b) is (1, 2a + 1). Find the value of a and b.

Solution 11

Mid-point of (2a, 4) and (-2, 2b) is (1, 2a + 1), therefore using mid-point formula, we have:

y =

2a + 1 =

a = 2

Putting, a = 2 in 2a + 1 = 2 + b, we get,

5 - 2 = b b = 3

Therefore, a = 2, b = 3.

Question 12

(i) Write down the co-ordinates of the point P that divides the line joining A (-4, 1) and B (17, 10) in the ratio 1: 2.

(ii) Calculate the distance OP, where O is the origin.

(iii) In what ratio does the y-axis divide the line AB?

Solution 12

(i) Co-ordinates of point P are

(ii) OP =

(iii) Let AB be divided by the point P (0, y) lying on y-axis in the ratio k: 1.

Thus, the ratio in which the y-axis divide the line AB is 4: 17.

Question 13

Prove that the points A (-5, 4), B (-1, -2) and C (5, 2) are the vertices of an isosceles right-angled triangle. Find the co-ordinates of D so that ABCD is a square.

Solution 13

We have:

AB = BC and

ABC is an isosceles right-angled triangle.

Let the coordinates of D be (x, y).

If ABCD is a square, then,

Mid-point of AC = Mid-point of BD

x = 1, y = 8

Thus, the co-ordinates of point D are (1, 8).

Question 14

M is the mid-point of the line segment joining the points A (-3, 7) and B (9, -1). Find the co-ordinates of point M. Further, if R (2, 2) divides the line segment joining M and the origin in the ratio p: q, find the ratio p: q.

Solution 14

Given, M is the mid-point of the line segment joining the points A (-3, 7) and B (9, -1).

The co-ordinates of point M are

Also, given that, R (2, 2) divides the line segment joining M and the origin in the ratio p: q.

Thus, the ratio p: q is 1: 2.

Question 15

Calculate the ratio in which the line joining A(-4, 2) and B(3, 6) is divided by point P(x, 3). Also, find

1. x
2. length of AP.
Solution 15

Question 16

Find the ratio in which the line 2x + y = 4 divides the line segment joining the points P(2, -2) and Q(3, 7).

Solution 16

Question 17

If the abscissa of a point P is 2. Find the ratio in which this point divides the line segment joining the point (-4, 3) and (6, 3). Al so, find the co-ordinates of point P.

Solution 17

Question 18

The line joining the points (2, 1) and (5, -8) is trisected at the points P and Q, point P lies on the line 2x - y + k = 0, find the value of k. Also, find the co-ordinates of point Q.

Solution 18

Question 19

Find the image of the point A(5, -3), under reflection in the point P(-1, 3).

Solution 19

Let A' = (x, y) be the image of the point A(5, -3), under reflection in the point P(-1, 3).

P(-1, 3) is the mid - point of the line segment AA'.

Therefore the image of the point A(5, -3), under reflection in the point P(-1, 3) is A'(-7, 9).

Question 20

M is the mid-point of the line segment joining the points A(0, 4) and B(6, 0). M also divides the line segment OP in the ratio 1 : 3. Find :

1. co-ordinates of M
2. co-ordinates of P
3. length of BP

Solution 20

Question 21

A(-4, 2), B(0, 2) and C(-2, -4) are the vertices of a triangle ABC. P, Q and R are mid-points of sides BC, CA and AB respectively. Show that the centroid of ∆ PQR is the same as the centroid of ∆ ABC.

Solution 21

Question 22

A(3, 1), B(y, 4) and C(1, x) are vertices of a triangle ABC. P, Q and R are mid - points of sides BC, CA and AB respectively. Show that the centroid of ΔPQR is the same as the centroid ΔABC.

Solution 22

P, Q and R are the mid points of the sides BC, CA and AB.

By mid - point formula, we get

From (i) and (ii), we get

Centroid of a ABC = Centroid of a PQR

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