Section Formula ML Aggarwal Solutions ICSE Class-10 Mathematics

Section Formula Chapter 11 ML Aggarwal Solution

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Section Formula ML Aggarwal Solutions ICSE Class-10 Mathematics Chapter-11 . We Provide Step by Step Answer of Exercise-11 Section Formula , with MCQs and Chapter-Test Questions  / Problems related  for ICSE Class-10 APC Understanding Mathematics  .  Visit official Website CISCE  for detail information about ICSE Board Class-10.

Section Formula ML Aggarwal Solutions ICSE Class-10 Maths Chapter-11


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ML Aggarwal Solutions Section Formula for ICSE Maths Chapter 11

Question 1

Find the co-ordinates of the mid-point of the line segments joining the following pairs of points:
(i) (2, – 3), ( – 6, 7)
(ii) (5, – 11), (4, 3)
(iii) (a + 3, 5b), (2a – 1, 3b + 4)

Answer 1

(i) Co-ordinates of the mid-point of (2, -3), ( -6, 7)
\left( \frac { { x }_{ 1 }+{ x }_{ 2 } }{ 2 } ,\frac { { y }_{ 1 }+{ y }_{ 2 } }{ 2 } \right) or
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 1

Question 2

 

The co-ordinates of two points A and B are ( – 3, 3) and (12, – 7) respectively. P is a point on the line segment AB such that AP : PB = 2 : 3. Find the co-ordinates of P.

Answer 2

Points are A (-3, 3), B (12, -7)
Let P (x1 y1) be the point which divides AB in the ratio of m1 : m2 i.e. 2 : 3
then co-ordinates of P will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 2.1

Question 3

P divides the distance between A ( – 2, 1) and B (1, 4) in the ratio of 2 : 1. Calculate the co-ordinates of the point P.

Answer 3

Points are A (-2, 1) and B (1, 4) and
Let P (x, y) divides AB in the ratio of m1 : m2 i.e. 2 : 1
Co-ordinates of P will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 3

Question 4

(i) Find the co-ordinates of the points of trisection of the line segment joining the point (3, – 3) and (6, 9).
(ii) The line segment joining the points (3, – 4) and (1, 2) is trisected at the points P and Q. If the coordinates of P and Q are (p, – 2) and \left( \frac { 5 }{ 3 } ,q \right)  respectively, find the values of p and q.

Answer 4

(i) Let P (x1, y1) and Q (x2, y2) be the points
which trisect the line segment joining the points
A (3, -3) and B (6, 9)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 4.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 4.2

Question 5

(i) The line segment joining the points A (3, 2) and B (5, 1) is divided at the point P in the ratio 1 : 2 and it lies on the line 3x – 18y + k = 0. Find the value of k.
(ii) A point P divides the line segment joining the points A (3, – 5) and B ( – 4, 8) such that \frac { AP }{ PB } =\frac { k }{ 1 }  If P lies on the line x + y = 0, then find the value of k.

Answer 5

(i) The point P (x, y) divides the line segment joining the points
A (3, 2) and B (5, 1) in the ratio 1 : 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 5.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 5.2

Question 6

Find the coordinates of the point which is three-fourth of the way from A (3, 1) to B ( – 2, 5).

Answer 6

Let P be the required point, then
\frac { AP }{ AB } =\frac { 3 }{ 4 }
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 6.1

Question 7

Point P (3, – 5) is reflected to P’ in the x- axis. Also P on reflection in the y-axis is mapped as P”.
(i) Find the co-ordinates of P’ and P”.
(ii) Compute the distance P’ P”.
(iii) Find the middle point of the line segment P’ P”.
(iv) On which co-ordinate axis does the middle point of the line segment P P” lie ?

Answer 7 

(i) Co-ordinates of P’, the image of P (3, -5)
when reflected in x-axis will be (3, 5)
and co-ordinates of P”, the image of P (3, -5)
when reflected in y-axis will be (-3, -5)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 7

Question 8

Use graph paper for this question. Take 1 cm = 1 unit on both axes. Plot the points A(3, 0) and B(0, 4).
(i) Write down the co-ordinates of A1, the reflection of A in the y-axis.

(ii) Write down the co-ordinates of B1, the reflection of B in the x-axis.

(iii) Assign.the special name to the quadrilateral ABA1B1.
(iv) If C is the mid point is AB. Write down the co-ordinates of the point C1, the reflection of C in the origin.
(v) Assign the special name to quadrilateral ABC1B1.

Answer 8

Two points A (3, 0) and B (0,4) have been plotted on the graph.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 8
(i)∵ A1 is the reflection of A (3, 0) in the v-axis Its co-ordinates will be ( -3, 0)
(ii)∵ B1 is the reflection of B (0, 4) in the .x-axis co-ordinates of B, will be (0, -4)
(iii) The so formed figure ABA1B1 is a rhombus.
(iv) C is the mid point of AB co-ordinates of C” will be \frac { AP }{ AB } =\frac { 3 }{ 4 }
∵ C, is the reflection of C in the origin
co-ordinates of C, will be \left( \frac { -3 }{ 2 } ,-2 \right)
(v) The name of quadrilateral ABC1B1 is a trapezium because AB is parallel to B1C1.

Question 9

The line segment joining A ( – 3, 1) and B (5, – 4) is a diameter of a circle whose centre is C. find the co-ordinates of the point C. (1990)

Answer 9

∵ C is the centre of the circle and AB is the diameter
C is the midpoint of AB.
Let co-ordinates of C (x, y)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 9

Question 10

The mid-point of the line segment joining the points (3m, 6) and ( – 4, 3n) is (1, 2m – 1). Find the values of m and n.


Answer 10

Let the mid-point of the line segment joining two points
A(3m, 6) and (-4, 3n) is P( 1, 2m – 1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 10

Question 11

The co-ordinates of the mid-point of the line segment PQ are (1, – 2). The co-ordinates of P are ( – 3, 2). Find the co-ordinates of Q.(1992)

Answer 11

Let the co-ordinates of Q be (x, y)
co-ordinates of P are (-3, 2) and mid-point of PQ are (1, -2) then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 11

Question 12

AB is a diameter of a circle with centre C ( – 2, 5). If point A is (3, – 7). Find:
(i) the length of radius AC.
(ii) the coordinates of B.

Answer 12

AC = \sqrt { { \left( 3+2 \right) }^{ 2 }+{ \left( -7-5 \right) }^{ 2 } }
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 12

Question 13

Find the reflection (image) of the point (5, – 3) in the point ( – 1, 3).

Answer 13

Let the co-ordinates of the images of the point A (5, -3) be
A1 (x, y) in the point (-1, 3) then
the point (-1, 3) will be the midpoint of AA1.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 13

Question 14

The line segment joining A \left( -1,\frac { 5 }{ 3 } \right)  the points B (a, 5) is divided in the ratio 1 : 3 at P, the point where the line segment AB intersects y-axis. Calculate
(i) the value of a
(ii) the co-ordinates of P. (1994)

Answer 14

Let P (x, y) divides the line segment joining
the points \left( -1,\frac { 5 }{ 3 } \right) , B(a, 5) in the ratio 1 : 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 14

Question 15

The point P ( – 4, 1) divides the line segment joining the points A (2, – 2) and B in the ratio of 3 : 5. Find the point B.

Answer 15

Let the co-ordinates of B be (x, y)
Co-ordinates of A (2, -2) and point P (-4, 1)
divides AB in the ratio of 3 : 5
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 15

Question 16

(i) In what ratio does the point (5, 4) divide the line segment joining the points (2, 1) and (7 ,6) ?
(ii) In what ratio does the point ( – 4, b) divide the line segment joining the points P (2, – 2), Q ( – 14, 6) ? Hence find the value of b.

Answer 16

(i) Let the ratio be m1 : m2 that the point (5, 4) divides
the line segment joining the points (2, 1), (7, 6).
5=\frac { { m }_{ 1 }\times 7+{ m }_{ 2 }\times 2 }{ { m }_{ 1 }+{ m }_{ 2 } }
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 16

Question 17

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

Answer 17


Let the co-ordinates of K be (x, 0) as it intersects x-axis.
Let point K divides the line segment joining the points
A (2, 3) and B (6, -5) in the ratio m1 : m2.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 17

Question 18

If A ( – 4, 3) and B (8, – 6), (i) find the length of AB.
(ii) in what ratio is the line joining AB, divided by the x-axis? (2008)

Answer 18

Given A (-4, 3), B (8, -6)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 18

Question 19

(i) Calculate the ratio in which the line segment joining (3, 4) and( – 2, 1) is divided by the y-axis.
(ii) In what ratio does the line x – y – 2 = 0 divide the line segment joining the points (3, – 1) and (8, 9)? Also, find the coordinates of the point of division.

Answer 19

(i) Let the point P divides the line segment joining the points

A (3, 4) and B (-2, 3) in the ratio of m1 : m2 and
let the co-ordinates of P be (0, y) as it intersects the y-axis
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 19
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 19.1

Question 20

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.

Answer 20

(i) Let the y-axis divide AB in the ratio m : 1. So,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 20
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 20.1

Question 21

(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 ?

Answer 21

(i) Let co-ordinate of P be (x, y) which divides the line segment joining the points
A ( -4, 1) and B(17, 10) in the ratio of 1 : 2.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 21

Question 22

Calculate the length of the median through the vertex A of the triangle ABC with vertices A (7, – 3), B (5, 3) and C (3, – 1)

Answer 22

Let D (x, y) be the median of ΔABC through A to BC.
∴ D will be the midpoint of BC
∴ Co-ordinates of D will be,
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 22

Question 23

Three consecutive vertices of a parallelogram ABCD are A (1, 2), B (1, 0) and C (4, 0). Find the fourth vertex D.

Answer 23

Let O in the mid-point of AC the diagonal of ABCD
∴ Co-ordinates of O will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 23

Question 24

If the points A ( – 2, – 1), B (1, 0), C (p, 3) and D (1, q) from a parallelogram ABCD, find the values of p and q.

Answer 24

A (-2, -1), B (1, 0), C (p, 3) and D (1, q)
are the vertices of a parallelogram ABCD
∴ Diagonal AC and BD bisect each other at O
O is the midpoint of AC as well as BD
Let co-ordinates of O be (x, y)
When O is mid-point of AC, then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 24
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 24.1

Question 25

If two vertices of a parallelogram are (3, 2) ( – 1, 0) and its diagonals meet at (2, – 5), find the other two vertices of the parallelogram.

Answer 25

Two vertices of a ||gm ABCD are A (3, 2), B (-1, 0)
and point of intersection of its diagonals is P (2, -5)
P is mid-point of AC and BD.
Let co-ordinates of C be (x, y), then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 25

Question 26

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.

Answer 26

Points A (-5, 4), B (-1, -2) and C (5, 2) are given.
If these are vertices of an isosceles triangle ABC then
AB = BC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 26
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 26.1

Question 27

Find the third vertex of a triangle if its two vertices are ( – 1, 4) and (5, 2) and mid point of one sides is (0, 3).

Answer 27

Let A (-1, 4) and B (5, 2) be the two points and let D (0, 3)
be its the midpoint of AC and co-ordinates of C be (x, y).
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 27
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 27.1

Question 28

Find the coordinates of the vertices of the triangle the middle points of whose sides are \left( 0,\frac { 1 }{ 2 } \right) ,\left( \frac { 1 }{ 2 } ,\frac { 1 }{ 2 } \right) and\left( \frac { 1 }{ 2 } ,0 \right)

Answer 28

Let ABC be a ∆ in which D\left( 0,\frac { 1 }{ 2 } \right) ,E\left( \frac { 1 }{ 2 } ,\frac { 1 }{ 2 } \right) andF\left( \frac { 1 }{ 2 } ,0 \right) ,
the mid-points of sides AB, BC and CA respectively.
Let co-ordinates of A be (x1, y1), B (x2, y2), C (x3, y3)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 28
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 28.1

Question 29

Show by section formula that the points (3, – 2), (5, 2) and (8, 8) are collinear.

Answer 29

Let the point (5, 2) divides the line joining the points (3, -2) and (8, 8)
in the ratio of m1 : m2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 29
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 29.1

Question 30

Find the value of p for which the points ( – 5, 1), (1, p) and (4, – 2) are collinear.

Answer 30

Let points A (-5, 1), B (1, p) and C (4, -2)
are collinear and let point A (-5, 1) divides
BC in the ratio in m1 : m2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 30

Question 31

A (10, 5), B (6, – 3) and C (2, 1) are the vertices of triangle ABC. L is the mid point of AB, M is the mid-point of AC. Write down the co-ordinates of L and M. Show that LM = \\ \frac { 1 }{ 2 }  BC.

Answer 31

Co-ordinates of L will be
\left( \frac { 10+6 }{ 2 } ,\frac { 5-3 }{ 2 } \right) or\left( \frac { 16 }{ 2 } ,\frac { 2 }{ 2 } \right) or(8,1)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 31
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 31.1

Question 32

A (2, 5), B ( – 1, 2) and C (5, 8) are the vertices of a triangle ABC. P and.Q are points on AB and AC respectively such that AP : PB = AQ : QC = 1 : 2.
(i) Find the co-ordinates of P and Q.
(ii) Show that PQ = \\ \frac { 1 }{ 3 }  BC.

Answer 32

 

A (2, 5), B (-1, 2) and C (5, 8) are the vertices of a ∆ABC,
P and Q are points on AB
and AC respectively such that \frac { AP }{ PB } =\frac { AQ }{ QC } =\frac { 1 }{ 2 }
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 32
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 32.1

 

Question 33

The mid-point of the line segment AB shown in the adjoining diagram is (4, – 3). Write down die co-ordinates of A and B.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 33

Answer 33

A lies on x-axis and B on the y-axis.
Let co-ordinates of A be (x, 0) and of B be (0, y)
P (4, -3) is the mid-point of AB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 33.1

Question 34

Find the co-ordinates of the centroid of a triangle whose vertices are A ( – 1, 3), B(1, – 1) and C (5, 1) (2006)

Answer 34

Co-ordinates of the centroid of a triangle,
whose vertices are (x1, y1), (x2, y2) and
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 34

Question 35

Two vertices of a triangle are (3, – 5) and ( – 7, 4). Find the third vertex given that the centroid is (2, – 1).

Answer 35

Let the co-ordinates of third vertices be (x, y)
and other two vertices are (3, -5) and (-7, 4)
and centroid = (2, -1).
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 35

Question 36

The vertices of a triangle are A ( – 5, 3), B (p – 1) and C (6, q). Find the values of p and q if the centroid of the triangle ABC is the point (1, – 1).

Answer 36

The vertices of ∆ABC are A (-5, 3), B (p, -1), C (6, q)
and the centroid of ∆ABC is O (1, -1)
co-ordinates of the centroid of ∆ABC will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Ex 11 36


MCQ , Chapter – 11  Section Formula Solutions of ML Aggarwal Maths for ICSE Class 10

Choose the correct answer from the given four options (1 to 12) :

Question 1

 The points A (9, 0), B (9, 6), C ( – 9, 6) and D ( – 9, 0) are the vertices of a

(a) rectangle
(b) square
(c) rhombus
(d) trapezium

Answer 1

A (9, 0), B (9, 6), C (-9, 6), D (-9, 0)
AB² = (x2 – x1)² + (y2 – y1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 1

Question 2

The mid-point of the line segment joining the points A ( – 2, 8) and B ( – 6, – 4) is
(a) ( – 4, – 6)
(b) (2, 6)
(c) ( – 4, 2)
(d) (4, 2)

Answer 2

Mid-point of the line segment joining the points A (-2, 8), B (-6, -4)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 2

Question 3

If P\left( \frac { a }{ 3 } ,4 \right)  segment joining the points Q ( – 6, 5) and R ( – 2, 3), then the value of a is
(a) – 4
(b) – 6
(c) 12
(d) – 12

Answer 3

P\left( \frac { a }{ 3 } ,4 \right)  is mid-point of the line segment
joining the points Q (-6, 5) and R (-2, 3)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 3

Question 4

If the end points of a diameter of a circle are A ( – 2, 3) and B (4, – 5), then the coordinates of its centre are
(a) (2, – 2)
(b) (1, – 1)
(c) ( – 1, 1)
(d) ( – 2, 2)

Answer 4

End points of a diameter of a circle are (-2, 3) and B (4,-5)
then co-ordinates of the centre of the circle
\left( \frac { -2+4 }{ 2 } ,\frac { 3-5 }{ 2 } \right) or\left( \frac { 2 }{ 2 } ,\frac { -2 }{ 2 } \right)
= (1, -1) (b)

Question 5

If one end of a diameter of a circle is (2, 3) and the centre is ( – 2, 5), then the other end is
(a) ( – 6, 7)
(b) (6, – 7)
(c) (0, 8)
(d) (0, 4)

Answer 5

One end of a diameter of a circle is (2, 3) and centre is (-2, 5)
Let (x, y) be the other end of the diameter
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 5

Question 6

If the mid-point of the line segment joining the points P (a, b – 2) and Q ( – 2, 4) is R (2, – 3), then the values of a and b are
(a) a = 4, b = – 5
(b) a = 6, b = 8
(c) a = 6, b = – 8
(d) a = – 6, b = 8

Answer 6

the mid-point of the line segment joining the
points P (a, b – 2) and Q (-2, 4) is R (2, -3)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 6
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 6.1

Question 7

The point which lies on the perpendicular bisector of the line segment joining the points A ( – 2, – 5) and B (2, 5) is
(a) (0, 0)
(b) (0, 2)
(c) (2, 0)
(d) ( – 2, 0)

Answer 7

the line segment joining the points A (-2, -5) and B (2, -5), has mid-point
\left( \frac { -2+2 }{ 2 } ,\frac { -5+5 }{ 2 } \right)  = (0, 0)
(0, 0) lies on the perpendicular bisector of AB. (a)

Question 8

The coordinates of the point which is equidistant from the three vertices of ∆AOB (shown in the given figure) are
(a) (x, y)
(b) (y, x)
(c) \left( \frac { x }{ 2 } ,\frac { y }{ 2 } \right)
(d) \left( \frac { y }{ 2 } ,\frac { x }{ 2 } \right)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 8

Answer 8

In the given figure, vertices of a ∆OAB are (0, 0), (0, 2y) and (2x, 0)
The point which is equidistant from O, A and B is the mid-point of AB.
∴ Coordinates are \left( \frac { 0+2x }{ 2 } ,\frac { 2y+0 }{ 2 } \right)  or (x, y) (a)

Question 9

The fourth vertex D of a parallelogram ABCD whose three vertices are A ( – 2, 3), B (6, 7) and C (8, 3) is
(a) (0, 1)
(b) (0, – 1)
(c) ( – 1, 0)
(d) (1, 0)

Answer 9

ABCD is a ||gm whose vertices A (-2, 3), B (6, 7) and C (8, 3).
The fourth vertex D will be the point on which diagonals AC and BD
bisect each other at O.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 9

 

Question 10

A line intersects the y-axis and x-axis at the points P and Q respectively. If (2, – 5) is the mid-point of PQ, then the coordinates of P and Q are, respectively
(a) (0, – 5) and (2, 0)
(b) (0, 10) and ( – 4, 0)
(c) (0, 4) and ( – 10, 0)
(d) (0, – 10) and (4, 0)

Answer 10

A line intersects y-axis at P and x-axis a Q.
R (2, -5) is the mid-point
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 10

 

Question 11

The points which divides the line segment joining the points (7, – 6) and (3, 4) in the ratio 1 : 2 internally lies in the
(a) Ist quadrant
(b) IInd quadrant
(c) IIIrd quadrant
(d) IVth quadrant

Answer 11

A point divides line segment joining the points
A (7, -6) and B (3, 4) in the ratio 1 : 2 internally.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 11
Let (x, y) divides it in the ratio 1 : 2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula MCQS 11.1

Question 12
The centroid of the triangle whose vertices are (3, – 7), ( – 8, 6) and (5, 10) is
(a) (0, 9)
(b) (0, 3)
(c) (1, 3)
(d) (3, 3)

Answer 12

Centroid of the triangle whose Vertices are (3, -7), (-8, 6) and (5, 10) is
\left( \frac { 3-8+5 }{ 3 } ,\frac { -7+6+10 }{ 3 } \right) or\left( 0,\frac { 9 }{ 3 } \right)
or (0, 3) (b)


Chapter Test ML Aggarwal Solutions Section Formula for ICSE Maths Chapter 11

Question 1

The base BC of an equilateral triangle ABC lies on y-axis. The coordinates of the point C are (0, – 3). If origin is the mid-point of the base BC, find the coordinates of the points A and B

Answer 1

Base BC of an equilateral ∆ABC lies on y-axis
co-ordinates of point C are (0, – 3),
origin (0, 0) is the mid-point of BC.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 1.1

Question 2

A and B have co-ordinates (4, 3) and (0, 1), Find
(i) the image A’ of A under reflection in the y – axis.
(ii) the image of B’ of B under reflection in the lineAA’.
(iii) the length of A’B’.

 

Answer 2

ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 2
(i) Co-ordinates of A’, the image of A (4, 3)
reflected in y-axis will be ( – 4, 3).
(ii) Co-ordinates of B’ the image of B (0, 1)
reflected in the line AA’ will be (0, 5).
(iii) Length A’B’
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 2.1

Question 3

Find the co-ordinates of the point that divides the line segment joining the points P (5, – 2) and Q (9, 6) internally in the ratio of 3 : 1.

Answer 3

Let R be the point whose co-ordinates are (x, y)
which divides PQ in the ratio of 3:1.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 3

Question 4

Find the coordinates of the point P which is three-fourth of the way from A (3, 1) to B ( – 2, 5).

Answer 4

Co-ordinates of A (3, 1) and B ( – 2, 5)
P lies on AB such that
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 4
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 4.1

Question 5

P and Q are the points on the line segment joining the points A (3, – 1) and B ( – 6, 5) such that AP = PQ = QB. Find the co-ordinates of P and Q.

Answer 5

Given
AP = PQ = QB
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 5

Question 6

The centre of a circle is (α + 2, α – 5). Find the value of a given that the circle passes through the points (2, – 2) and (8, – 2).

Answer 6

Let A (2, -2), B (8, -2) and centre of the circle be
O (α + 2, α – 5)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 6

Question 7

The mid-point of the line joining A (2, p) and B (q, 4) is (3, 5). Calculate the numerical values of p and q.

Answer 7

Given
(3, 5) is the mid-point of A (2, p) and B (q, 4)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 7

Question 8

The ends of a diameter of a circle have the co-ordinates (3, 0) and ( – 5, 6). PQ is another diameter where Q has the coordinates ( – 1, – 2). Find the co-ordinates of P and the radius of the circle.

Answer 8

Let AB be the diameter where co-ordinates of
A are (3, 0) and of B are (-5, 6).
Co-ordinates of its origin O will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 8

Question 9

In what ratio does the point ( – 4, 6) divide the line segment joining the points A( – 6, 10) and B (3, – 8) ?

Answer 9

Let the point (-4, 6) divides the line segment joining the points
A (-6, 10) and B (3, -8), in the ratio m : n
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 9

Question 10

Find the ratio in which the point P ( – 3, p) divides the line segment joining the points ( – 5, – 4) and ( – 2, 3). Hence find the value of p.

Answer 10

Let P (-3, p) divides AB in the ratio of m1 : m2 coordinates of
A (-5, -4) and B (-2, 3)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 10

Question 11

In what ratio is the line joining the points (4, 2) and (3, – 5) divided by the x-axis? Also find the co-ordinates of the point of division.

Answer 11

Let the point P which is on the x-axis, divides the line segment
joining the points A (4, 2) and B (3, -5) in the ratio of m1 : m2.
and let co-ordinates of P be (x, 0)
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 11
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 11.1

Question 12

If the abscissa of a point P is 2, find the ratio in which it divides the line segment joining the points ( – 4 – 3) and (6, 3). Hence, find the co-ordinates of P.

Answer 12

Let co-ordinates of A be (-4, 3) and of B (6, 3) and of P be (2, y)
Let the ratio in which the P divides AB be m1 : m2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 12

Question 13

Determine the ratio in which the line 2x + y – 4 = 0 divide the line segment joining the points A (2, – 2) and B (3, 7). Also find the co-ordinates of the point of division.

Answer 13

Points are given A (2, -2), B (3, 7)
and let the line 2x + y – 4 = 0 divides AB in the ratio m1 : m2
at P and let co-ordinates of
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 13
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 13.1

Question 14

The point A(2, – 3) is reflected in the v-axis onto the point A’. Then the point A’ is reflected in the line x = 4 onto the:point A”.
(i) Write the coordinates of A’ and A”.
(ii) Find the ratio in which the line segment AA” is divided by the x-axis. Also find the coordinates of the point of division.

Answer 14

A’ is the reflection of A(2, -3) in the x-axis
(i) ∴ Co-ordinates of A’ will be (2, 3)
Draw a line x = 4 which is parallel to y-axis
A” is the reflection of A’ (2, 3)
∴Co-ordinates OA” will be (6, 3)
(ii) Join AA” which intersects x-axis at P whose
co-ordinate are (4, 0)
Let P divide AA” in the ratio in m1 : m2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 14
Hence P(4, 0) divides AA” in the ratio 1 : 1

Question 15

ABCD is a parallelogram. If the coordinates of A, B and D are (10, – 6), (2, – 6) and (4, – 2) respectively, find the co-ordinates of C.

Answer 15

Let the co-ordinates of C be (x, y) and other three vertices
of the given parallelogram are A (10, – 6), B, (2, – 6) and D (4, – 2)
∴ ABCD is a parallelogram
Its diagonals bisect each other.
Let AC and BD intersect each other at O.
∴O is mid-points of BD
∴ Co-ordinates of O will be
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 15

Question 16

ABCD is a parallelogram whose vertices A and B have co-ordinates (2, – 3) and ( – 1, – 1) respectively. If the diagonals of the parallelogram meet at the point M(1, – 4), find the co-ordinates of C and D. Hence, find the perimeter of the parallelogram. find the perimeter of the parallelogram.

Answer 16

ABCD is a || gm , m which co-ordinates of A are (2, -3) and B (-1, -1)
Its diagonals AC and BD bisect each other at M (1, -4)
∴ M is the midpoint of AC and BD
Let co-ordinates of C be (x1, y1) and of D be (x2, y2)
when M is the midpoint of AC then
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 16
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 16.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 16.2

Question 17

In the adjoining figure, P (3, 1) is the point on the line segment AB such that AP : PB = 2 : 3. Find the co-ordinates of A and B.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 17

Answer 17

A lies on x-axis and
B lies on y-axis
Let co-ordinates of A be (x, 0) and B be (0, y)
and P (3, 1) divides it in the ratio of 2 : 3.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 17.1

Question 18

Given, O, (0, 0), P(1, 2), S( – 3, 0) P divides OQ in the ratio of 2 : 3 and OPRS is a parallelogram.
Find : 

(i) the co-ordinates of Q.
(ii)the co-ordinates of R.
(iii) the ratio in which RQ is divided by y-axis.
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 18

 

Answer 18

(i) Let co-ordinates of Q be (x’, y’) and of R (x”, y”)
Point P (1, 2) divides OQ in the ratio of 2 : 3
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 18.1
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 18.2
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 18.3

Question 19

If A (5, – 1), B ( – 3, – 2) and C ( – 1, 8) are the vertices of a triangle ABC, find the length of the median through A and the co-ordinates of the centroid of triangle ABC.

Answer 19

A (5, -1), B (-3, -2) and C (-1, 8) are the vertices of ∆ABC
D, E and F are the midpoints of sides BC, CA and AB respectively
and G is the centroid of the ∆ABC
ML Aggarwal Class 10 Solutions for ICSE Maths Chapter 11 Section Formula Chapter Test 19

— : End of Section Formula ML Aggarwal Solutions :–


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