# Chapter 13 : Section and Mid-Point Formula - Selina Solutions for Class 10 Maths ICSE

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

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.

(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.

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

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.

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

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.

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

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

Using section formula, we have:

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

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:

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.

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 .

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.

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:

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.

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:

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.

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

** **

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.

5AP = 2BP

The co-ordinates of the point P are

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.

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).

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

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.

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.

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).

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

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

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

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.

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.

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

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?

(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.

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.

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.

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.

(i) Co-ordinates of P are

Co-ordinates of Q are

(ii) Using distance formula, we have:

BC =

PQ =

Hence, PQ = BC.

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.

BP: PC = 2: 3

Co-ordinates of P are

Using distance formula, we have:

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.

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 .

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.

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 .

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.

(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 .

(iii) Area of quadrilateral PMNQ

= (PM + QN) MN

= (5 + 2) 7

= 7 7

= 24.5 sq units

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.

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.

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

(i)

(ii)

(iii)

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.

Take (x_{1}
, y_{1}) = (-3, 3a +
1) ; (x_{2}
, y_{2}) = B(5, 8a)
and

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

Here m_{1} = 3 and m_{2} =1

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

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

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

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

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

Mid-point of AB =

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

Mid-point of AB =

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.

Mid-point of AB = (2, 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.

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:

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).

(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).

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.

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.

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

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.

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

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,

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

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).

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).

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).

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.

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.

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.

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).

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

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).

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

Let A(x_{1},y_{1}), 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).

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.

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

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.

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.

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

Co-ordinates of the centroid of triangle ABC are

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.

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)

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.

Given, centroid of triangle ABC is the origin.

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

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.

Given, BP: PC = 3: 2

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

Using distance formula, we have:

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.

Using section formula,

Given, AB = 6AQ

Using section formula,

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.

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.

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

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).

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'.

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.

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.

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).

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

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).

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.

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

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

Co- ordinates of the centroid of triangle ABC are

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

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

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

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.

(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?

(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.

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.

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).

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.

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.

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

- x
- length of AP.

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

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.

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.

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

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).

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 :

- co-ordinates of M
- co-ordinates of P
- length of BP

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.

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.

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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