# Section Formula ML Aggarwal Solutions ICSE Class-10 Mathematics

## Section Formula Chapter 11 ML Aggarwal Solution

**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)

**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 (x_{1}, _{ }y_{1}) be the point which divides AB in the ratio of m_{1} : m_{2} i.e. 2 : 3

then co-ordinates of P will be

**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 m_{1} : m_{2} i.e. 2 : 1

Co-ordinates of P will be

**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 respectively, find the values of p and q.**

**Answer 4**

(i) Let P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) be the points

which trisect the line segment joining the points

A (3, -3) and B (6, 9)

**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 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

**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

**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)

**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.

(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

∵ C, is the reflection of C in the origin

co-ordinates of C, will be

(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)

**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

Answer 10

Let the mid-point of the line segment joining two points

A(3m, 6) and (-4, 3n) is P( 1, 2m – 1)

**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

**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 =

**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.

**Question 14**

**The line segment joining A 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 , B(a, 5) in the ratio 1 : 3

**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

**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 m_{1} : m_{2} that the point (5, 4) divides

the line segment joining the points (2, 1), (7, 6).

**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 m

_{1}: m

_{2}.

**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)

**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 m_{1} : m_{2} and

let the co-ordinates of P be (0, y) as it intersects the y-axis

**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,

**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.

**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,

**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

**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

**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

**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.

**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).

**Question 28**

**Find the coordinates of the vertices of the triangle the middle points of whose sides are **

**Answer 28**

Let ABC be a ∆ in which ,

the mid-points of sides AB, BC and CA respectively.

Let co-ordinates of A be (x_{1}, y_{1}), B (x_{2}, y_{2}), C (x_{3}, y_{3})

**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 m_{1} : m_{2}

**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 m_{1} : m_{2}

**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 = BC.**

**Answer 31**

Co-ordinates of L will be

**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 = 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

**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.**

**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

**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

**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).

**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

**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 = (x_{2} – x_{1})² + (y_{2} – y_{1})²

**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)

**Question 3**

**If 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**

is mid-point of the line segment

joining the points Q (-6, 5) and R (-2, 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

=

= (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

**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)

**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

= = (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) **

**(d) **

**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 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.

**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

**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.

Let (x, y) divides it in the ratio 1 : 2

**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

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.

**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**

(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’

**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.

**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

**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

**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)

**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)

**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

**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

**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 m_{1} : m_{2} coordinates of

A (-5, -4) and B (-2, 3)

**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 m_{1} : m_{2}.

and let co-ordinates of P be (x, 0)

**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 m_{1} : m_{2}

**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 m_{1} : m_{2}

at P and let co-ordinates of

**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 m_{1} : m_{2}

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

**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 (x_{1}, y_{1}) and of D be (x_{2}, y_{2})

when M is the midpoint of AC then

**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.**

**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.

**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.**

**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

**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