Assignments in Mathematics Class X (Term II)

7. COORDINATE GEOMETRY
IMPORTANT TERMS, DEFINITIONS AND RESULTS

• In the rectangular coordinate system, two number and the diagonals are also equal.
lines are drawn at right angles to each other. The (iv) parallelogram, prove that the opposite sides are
point of intersection of these two number lines is equal.
called the origin whose coordinates are taken as (v) parallelogram but not a rectangle, prove that
(0, 0). The horizontal number line is known as the its opposite sides are equal but diagonals are
x-axis and the vertical one as the y-axis. not equal.
• In the ordered pair (p, q), p is called the x-coordinate • Three points A, B and C are said to be collinear,
or abscissa and q is known as y-coordinate or if they lie on the same straight line.
ordinate of the point. • For three points to be collinear, the sum of the
• The coordinate plane is divided into four distances between two pairs of points is equal to
quadrants. the third pair of points.

N
• Three points will make :

A
(i) a scalene triangle, if no two sides of the

SH
triangle are equal.
(ii) an isosceles triangle, if any two sides are

A
equal.

K
(iii) an equilateral triangle, if all the three sides
are equal.

A
PR (iv) a right triangle, if sum of the squares of any
two sides is equal to the square of the third
• The abscissa of a point is its perpendicular distance
side.
from y-axis.
• • The coordinates of the point P(x, y) which divides
S
The ordinate of a point is its perpendicular distance
from x-axis. the line segment joining A(x1, y1) and B(x2, y2)
ER

• The abscissa of every point situated on the right side internally in the ratio m : n, are given by :
of y-axis is positive and the abscissa of every point mx2 + nx1 my2 + ny1
TH

situated on the left side of y-axis is negative. x= , y=

m+n m+n .
• The ordinate of every point situated above x-axis
is positive and that of every point below x-axis • The coordinates of the mid-point M of a line segment
O

is negative. AB with end points A(x1, y1) and B(x2, y2) are :
R

• The abscissa of every point on y-axis is zero.  x1 + x2 , y1 + y2 

•
B

The ordinate of every point on x-axis is zero.  2 2 

• The distance between any two points P(x1, y1) and
• The point of intersection of the medians of a triangle
L

Q (x2, y2) is given by

is called its centroid.
YA

2 2
PQ = (x2 − x1 ) + ( y2 − y1 ) • The coordiantes of the centroid of the triangle
whose vertices are (x1, y1), (x2, y2) and (x3, y3) are
O

2 2
or PQ = (x1 − x2 ) + ( y1 − y2 ) given by
 x1 + x2 + x3 , y1 + y2 + y3 
G

⇒ PQ =

 3 3  .
2 2
(Differenceof absissae) + (Differenceof ordinates)
• If O(0, 0) is the origin and P(x, y) is any point,
• The area of a DABC with vertices A(x 1, y1),
B(x2, y2) and C(x3, y3) is given by :
then from the above formula, we have :
2 2 2 2 area (DABC) =
(x - 0) + ( y - 0) = x + y 1
OP =  {x1 ( y2 − y3 ) + x2 ( y3 − y 1 ) + x3 ( y1 − y2 )} .
•In order to prove that a given figure is a : 2
(i) square, prove that four sides are equal and the Since area of a triangle cannot be negative, we
diagonals are equal. consider the absolute or numerical value of the
area.
(ii) rhombus, prove that the four sides are equal.
(iii) rectangle, prove the opposite sides are equal • Three given points A(x 1 , y 1 ), B(x 2 , y 2 ) and
C(x3, y3), are collinear if
1
 ⇔ area of DABC = 0 ⇔ x1 (y2 – y3) + x2(y3 – y1) + x3(y1 – y2) = 0
1
⇔ [x1 (y2 – y3) + x2(y3 – y1) + x3(y1 – y2)] = 0
2

Summative Assessment

MULTIPLE CHOICE QUESTIONS [1 Mark]

A. Important Questions

1. Three points A, B, C are said to be collinear, 11. The distance between the points P(2, –3) and
if : Q(2, 2) is :
(a) they lie on the same straight line (a) 2 units (b) 3 units

N
(b) they do not lie on the same straight line (c) 4 units (d) 5 units

A
(c) they lie on three different straight lines 12. If the points P(2, 3), Q(5, k) and R(6, 7) are

SH
(d) none of these collinear, then the value of k is :

2. The point (–3, 5) lies in : 3 1

(a) 4 (b) 6 (c) −(d)

A
(a) 1st quadrant (b) 2nd quadrant 2 4
13. The points M(0, 6), N(–5, 3) and P(3, 1) are the

K
(c) 3rd quadrant (d) 4th quadrant vertices of a triangle, which is :

A
3. The points A(0, –2), B(3, 1), C(0, 4) and (a) isosceles (b) equilateral
D(–3, 1) are the vertices of a : PR
(c) scalene (d) right angled
(a) parallelogram (b) rectangle
14. A is a point on y-axis at a distance of 4 units
(c) square (d) rhombus
S
from x-axis lying below x-axis. The coordinates
of A are :
ER

4. S is a point on x-axis at a distance of 4 units from

y-axis to its right. The coordinates of S are : (a) (4, 0) (b) (0, 4)
(a) (4, 0) (b) (0, 4)
TH

(c) (– 4, 0) (d) (0, – 4)

(c) (4, 4) (d) (– 4, 4)
15. The mid-point of the line segment joining the
5. The distance between the points P(0, y) and
O

points A(–2, 8) and B(–6, –4) is :

Q(x, 0) is given by :
(a) (–6, – 4) (b) (2, 6)
R

(a) x2 + y2 (b) x2 − y 2
(c) (– 4, 2) (d) (4, 2)
B

(c) x2 + y 2 (d) xy
16. The point which divides the line segment joining
L

6. The distance of the point P(2, 3) from the x-axis the points (7, –6) and (3, 4) in the ratio 1 : 2
is :
YA

internally lies in the :

(a) 2 (b) 3 (c) 1 (d) 5
(a) 1st quadrant (b) 2nd quadrant
O

7. The distance of the point P(–6, 8) from the origin

(c) 3rd quadrant (d) 4th quadrant
is :
G

(a) 8 (b) 2 7 (c) 10 (d) 6 17. If the point P(2, 1) lies on the line segment joining
points A(4, 2) and B(8, 4), then :
8. If the distance between the points (2, –2) and (–1, AB
x) is 5, one of the values of x is : (a) AP = (b) AP = AB
3
(a) – 2 (b) 2 (c) –1 (d) 1
AB AB
9. The distance between the points (0, 5) and (5, 0) (c) PB = (d) AP =
is : 3 2
(a) 5 (b) 5 2 (c) 2 5 (d) 10 18. If the points (k, 2k), (3k, 3k) and (3, 1) are collinear,
then k is :
10. The point on the x-axis which is equidistant from 2
1 1 2
P(–2, 9) and Q (2, –5) is : (a) (b) – (c) (d) –
3 3 3 3
(a) (0, 7) (b) (–7, 0)
(c) (7, 0) (d) (7, –7)
2
 19. The points (0, 6), (–5, 3) and (3, 1) are the vertices 30. If the distance between the points (4, p) and
of a triangle which is : (1, 0) is 5, then the value of p is :
(a) equilateral (b) isosceles (a) 4 only (b) ±4
(c) scalene (d) right angled (c) – 4 only (d) 0
 a 
20. If P  , 4  is the mid-point of the line segment 31. A line intersects x and y-axes at P and Q
3 respectively. If (2, –5) is the mid-point of PQ, then
joining the point Q(–6, 5) and R(–2, 3), then the the coordinates of P and Q are respectively:
value of a is :
(a) – 4 (b) –12 (c) 12 (d) –6 (a) (4, 0) and (0, –10) (b) (2, 0) and (0, –5)

21. The perpendicular bisector of the line segment (c) (– 4, 0) and (0, 10) (d) (–10, 0) and (0, 4)
joining the points A(1, 5) and B(4, 6) cuts the
32. The distance between the points (cos θ , sin θ ) and
y-axis at : (sin θ , cos θ ) is :
(a) (0, 13) (b) (0, –13)
(c) (0, 12) (d) (13, 0) (a) 3 (b) 2 (c) 2 (d) 1

N
22. The ratio in which (4, 5) divides the join of 33. If the points (1, 2), (– 5, 6) and (a, – 2) are collinear,

A
(2, 3) and (7, 8) is : then a is equal to :

SH
(a) 4 : 3 (b) 5 : 2 (a) –3 (b) 7 (c) 2 (d) – 2
(c) 3 : 2 (d) 2 : 3 34. The points A(9, 0), B(9, 6), C(– 9, 6) and

A
23. The y-axis divides the join of P(–4, 2) and D(– 9, 0) are the vertices of a :

K
Q(8, 3) in the ratio : (a) square (b) rectangle

A
(a) 3 : 1 (b) 1 : 3 (c) rhombus (d) trapezium
(c) 2 : 1 (d) 1 : 2 PR
35. AOBC is a rectangle whose three vertices are
24. Two vertices of DPQR are P(–1, 4) and Q(5, 2) A(0, 3), O(0, 0) and B(5, 0). The length of its
and its centroid is G(0, –3). The coordinates of diagonal is :
S
R are: (a) 5 (b) 3 (c) 34 (d) 4
ER

(a) (4, 3) (b) (4, 15) 36. The points (– 4, 0), (4, 0) and (0, 3) are vertices
(c) (–4, –15) (d) (–15, – 4) of a:
25. The x-axis divides the join of A(2, –3) and
TH

(a) right triangle (b) isosceles triangle

B(5, 6) in the ratio : (c) equilateral triangle (d) scalene triangle
(a) 1 : 2 (b) 2 : 1
O

37. The coordinates of the vertices of an equilateral

(c) 3 : 5 (d) 2 : 3 triangle are A(3, y), B(3, 3 ) and C(0, 0). The
R

26. A point A divides the join of X(5, –2) and value of y is :

B

Y(9, 6) in the ratio 3 : 1. The coordinates of A (a) 4 (b) 5

are : (c) –1 (d) none of these
L

(a) (4, 7) (b) (8, 4)

YA

38. The distance between the points (a cos q + b

 11  sin q, 0) and (0, a sin q – b cos q) is :
(c)  , 5  (d) (12, 8)
2  (a) a2 + b2 (b) a + b
O

27. If M(–1, 1) is the mid-point of the line segment

G

(c) a2 – b2 (d) a 2 + b2
joining P(–3, y) and Q(1, y + 4), then the value
of y is : 39. (–1, 2), (2, –1) and (3, 1) are three vertices of a
(a) 1 (b) –1 (c) 2 (d) 0 parallelogram. The coordinates of the fourth vertex
are :
28. If (x, 2), (–3, –4) and (7, –5) are collinear, then
x is equal to : (a) (4, 0) (b) (– 2, 0)
(a) 60 (b) 63 (c) – 63 (d) – 60 (c) (– 2, 6) (d) (6, 2)
29. If the area of the triangle formed by the points 40. If A(5, 3), B(11, –5) and C(12, a) are the vertices
(a, 2a), (–2, 6) and (3, 1) is 5 square units, then of a right angled triangle, right angled at C, then
a is equal to : the value of a is :
3 (a) – 2, 4 (b) –2, –4
(a) 2 (b) (c) 3 (d) 5
5 (c) 2, –4 (d) 2, 4

3
 41. The perimeter of the triangle with vertices (0, 4), (c) (x + y + z)2 (d) 0
(0, 0) and (3, 0) is : 50. The fourth vertex D of a parallelogram ABCD
(a) 5 (b) 12 whose three vertices are A(–2, 3), B(6, 7) and
(c) 11 (d) 7 + 5 C(8, 3) is :
42. The point which lies on the perpendicular bisector (a) (0, 1) (b) (0, –1)
of the line segment joining the points A(– 2, –5) (c) (–1, 0) (d) (1, 0)
and B(2, 5) is :
51. If the points A(1, 2), B(0, 0) and C(a, b) are
(a) (0, 0) (b) (0, 2) collinear, then :
(c) (2, 0) (d) (– 2, 0) (a) a = b (b) a = 2b
43. The area of the triangle with vertices (a, b + c), (c) 2a = b (d) a = –b
(b, c + a) and (c, a + b) is :
52. In the figure, OAB is a triangle. The coordinates
(a) (a + b + c)2 (b) 0 of the point which is equidistant from the three
(c) (a + b + c) (d) abc vertices are :

N
44. If the centroid of the triangle formed by the points (a) (x, y)

A
(a, b), (b, c) and (c, a) is at the origin, then
(b) (y, x)

SH
a3 + b3 + c3 is equal to :
(a) abc (b) 0 x y
(c)  , 

A
(c) a + b + c (d) 3abc 2 2

K
45. The line segment joining points (–3, – 4) and  y x
(d)  , 

A
(1, – 2) is divided by y-axis in the ratio :  2 2
(a) 1 : 3 (b) 2 : 3 PR
(c) 3 : 1 (d) 2 : 3
46. If points (x, 0), (0, y) are (1, 1) and collinear, 53. A circle drawn with origin as the centre passes
S

1 1  13 
through  , 0 The point which does not lie in
ER

then + is equal to : 2 
x y
the interior of the circle is :
(a) 1 (b) 2 (c) 0 (d) –1
TH

 3  7
47. If the ratio in which P divides the line segment (a)  − , 1 (b)  2, 
joining (x1, y1) and (x2, y2) be k : 1, then the  4   3
O

coordinates of the point P are :  1 5


(c)  5,  (d)  −6,
R

2  
 kx1 + x2 ky2 + y1  2
B

(a)  , 
k +1 k +1  54. The coordinates of the centroid of the triangle
L

with vertices (a, 0), (0, b) and (a, b) are :

 kx2 + x1 ky2 + y1 
YA

(b)  ,   a b
k +1 k +1   a b
(a)  ,  (b)  , 
2 2 3 3
 x1 + x2 y1 + y2 
O

(c)  ,
 k + 1 k + 1   2a 2b 
(c)  ,  (d) none of these
G

 3 3
(d) none of these
55. Area of the triangle with vertices (x, 0), (0, y)
48. Area of a triangle is taken :
and (x, y) is :
(a) always positive
x xy
(b) always negative (a) x + y (b) x – y (c) (d)
y z
(c) sometimes positive sometimes negative
56. If the area of a quadrilateral ABCD is zero, then
(d) none of these the four points A, B, C and D are :
49. The area of the triangle formed by (x, y + z), (a) collinear (b) not collinear
(y, z + x) and (z, x + y) is : (c) nothing can be said (d) none of these
(a) x + y + z (b) xyz

4
 B. Questions From CBSE Examination Papers

1. If the points (0, 0), (1, 2) and (x, y) are collinear (a) 13 units (b) 5 units
then : [2011 (T-II] (c) 12 units (d) 17 units
(a) x = y (b) 2x = y 3. The perimeter of a triangle with vertices (0, 4),
(c) x = 2y (d) 2x = –y (0, 0) and (3, 0) is : [2011 (T-II]
2. The perpendicular distance of A (5, 12) from the (a) 8 (b) 10 (c) 12 (d) 15
y-axis is : [2011 (T-II]

Short Answer Type Questions [2 Marks]

A. Important Questions

1. Find the distance between the points ( a cos35º , 0) similar to DDEF with vertices D(– 4, 0), E(4, 0) and

N
and (0 , a cos 55º ) F(0, 4) ? Justify your answer.

A
2. Find the value of x which is an integer such 17. Find the points on the y-axis, each of which is
that the distance between the points P(x, 2) and at a distance of 13 units from the point (–5, 7).

SH
Q(3, – 6) is 10 units.
3. Is the point (4, 4) equidistant from the points 18. Are the points A(4, 5), B(7, 6) and C(6, 3)

A
P(–1, 4) and Q(1, 0)? collinear?

K
4. A is a point on the x-axis and B is a point on the 19. If the points P(a, –11), Q(5, b), R(2, 15) and
S(1, 1) are the vertices of a parallelogram PQRS,

A
y-axis. If the abscissa of A be a and the ordinate of
B be –a, then find the length of segment AB. find the values of a and b.
5. What is the distance between A on the x-axis
PR
20. Find all possible values of a for which the distance
whose abscissa is 11 and B(7, 3)? between the points A(a, –1) and B(5, 3) is 5
units.
S
6. Find the coordinates of the other end of a diameter
21. Show that the points A(3, 1), B(12, –2) and
ER

of a circle whose one end is A(2, 1) and centre

C(0, 2) cannot be the vertices of a triangle.
 3 −5 
is P  , .
 2 2 
TH

22. If the points A(– 6, 10), B(– 4, 6) and C(3, –8)

2
7. Find the coordinates of the centroid of the triangle are collinear, then show that AB = AC.
whose vertices are (0, 6), (8, 12) and (8, 0). 9
O

8. If the point (a, b) is equidistant from the points 23. Find the coordinates of the point R which divides
R

(7, 1) and (3, 5), find the relation between a and the line segment joining the points P(–2, 3) and
B

b. Q(4, 7) internally in the ratio 4 : 7.

9. Two vertices of a triangle are A(–7, 4) and 24. If the point C(–1, 2) divides internally the line
L

B(3, – 5). If its centroid is (2, –1), then find the segment joining A(2, 5) and B in the ratio 3 : 4,
YA

coordinates of the third vertex C. find the coordinates of B.

10. The mid-point of the line segment joining (3, 6) 25. Check whether the point P(–2, 4) lies on a circle
O

and (x, 2) is (2, y). Find the values of x and y. of radius 6 units and centre C(3, 5).
G

11. What point on the x-axis is equidistant from 26. Does the point A(– 4, 2) lie on the line segment
(7, 6) and (–3, 4) ? joining the points X(– 4, 6) and Y(– 4, –6) ?
12. Find the ratio in which the line segment joining Justify your answer.
the points (6, 4) and (1, –7) is divided by the 27. In what ratio does the point P(2, –5) divide the
x-axis. line segment joining A(– 3, 5) and B(4, –9) ?
13. F i n d t h e d i s t a n c e b e t w e e n t h e p o i n t s 28. Show that the points A(–1, –2), B(4, 3),
A(x + y, x – y) and B(x – y, –x – y). C(2, 5) and D(–3, 0) are the vertices of the
14. Are the points A(a, b + c), B(b, c + a) and rectangle ABCD.
C(c, a + b) collinear ? 29. Show that A(–2, 3), B(8, 3) and C(6, 7) are the
15. Find the value of x if the points (x, 8), (– 4, 2) vertices of a right angled triangle.
and (5, –1) are collinear. 30. Find the value of m if the point (0, 2) is equidistant
16. Is DABC with vertices A(– 2, 0), B(2, 0), C(0, 2) from (3, m) and (m, 5).
5
 B. Questions From CBSE Examination Papers
1. If the point A (4, 3) and B(x, 5) are on the circle 18. Find the value of p, for which the points (1, 3),
with centre O(2, 3), find the value of x. (3, p) and (5, –1) are collinear. [2011 (T-II)]
[2011 (T-II)]
19. Find points on the x-axis, which are at a distance
2. Three consecutive vertices of a parallelogram of 5 units from the point A (5, –3).[2011 (T-II)]
ABCD are A(1, 2), B(1, 0) and C(4, 0). Find the
foruth vertex D. [2011 (T-II)] 20. Show that the points (a, b + c), (b, c + a) and
(c, a + b) are collinear. [2011 (T-II)]
3. Which point on x-axis is equidistant from (7, 6)
and (–3, 4)? [2011 (T-II)] 21. Prove that the points (0, 0), (5, 5) and (–5, 5) are
the vertices of a right angled isosceles triangle.
4. Find the ratio in which the line 2x + y = 4 divides
the join of A(2, –2) and B(3, 7). Also, find the 22. Find the value of x, if the distance between the
coordinates of the point of their intersection. points (x, –1) and (3, –2) is x + 5. [2011 (T-II)]
[2011 (T-II)] 23. If the point C (–1, 2) divides internally the line
5. Find the value of k for which the distance between segment joining A (2, 5) and B(x, y) in the ratio

N
(9, 2) and (3, k) is 10 units. [2011 (T-II)] 3 : 4, then find the coordinates of B.

A
[2011 (T-II)]
 −2 −20 

SH
6. Find the ratio in which the point  ,  24. Show that the point P (–4, 2) lies on the line
7 7 
divides the join of (–2, –2) and (2, – 4). segment joining the points A (– 4, 6) and
B (– 4, –6). [2011 (T-II)]

A
[2011 (T-II)]
25. Show that the point (1, –1) is the centre of the

K
7. Find the coordinates of a point on x-axis which
divides the line segment joining the points circle circumscribing the triangle whose vertices

A
(–2, –3) and (1, 6) in the ratio 1 : 2. are (4, 3) and (–2, 3) and (6, –1). [2011 (T-II)]
[2011 (T-II)]
PR26 If A(1, 2), B (4, y), C(x, 6) and D(3, 5) are the
vertices of a parallelogram ABCD taken in order,
8. The line segment joining the points P(3, 3) and find the values of x and y. [2011 (T-II)]
S
Q(6, – 6) is trisected at the points A and B such
that A is nearer to P. If A also lies on the line 27. If the point P(x, y) is equidistant from the points
ER

given by 2x + y + k = 0, find the value of k. A(5, 1) and B (–1, 5) then prove that 3x = 2y.
[2011 (T-II)]
9. Find the type of triangle formed by points
TH

A(–5, 6), B(– 4, –2), C(7, 5). [2011 (T-II)] 28. Determine the ratio in which the point P(x, –2)
divides the join of A(–4, 3) and B(2, –4). Also
10. In what ratio does the point (– 4, 6) divide the find the value of x. [2011 (T-II)]
O

line segment joining the points A(– 6, 10) and

B(3, –8)? [2011 (T-II)] 29. If points A(–2, –1), B (a, 0), C (4, b) and
R

D(1, 2) are the vertices of a parallelogram ABCD,

11. Find a relation between x and y such that the
B

find the values of a and b. [2011 (T-II)]

point P(x, y) is equidistant from the points
A(7, 1) and B(3, 5). [2011 (T-II)] 30. Find the value of x such that PQ = QR, where
L

the coordinates of P, Q and R are (6, –1), (1, 3)

YA

12. If the points A(6, 1), B(8, 2), C(9, 4) and D(P, 3) and (x, 8) respectively. [2011 (T-II)]
are the vertices of a parallelogram taken in order,
find the value of P. [2011 (T-II)] 31. Find the value of k for which the point (8, 1),
O

(k, –4) and (2, –5) are collinear. [2011 (T-II)]

13. Find the co-ordinates of the points of trisection
G

of the line segment joining the points A(2, –2) 32. Show that P(1, –1) is the centre of the circle
circumscribing the triangle whose angular points
and B(–7, 4). [2011 (T-II)]
are A (4, 3), B(–2, 3) and C (6, –1).
14. Check whether the points (2, 2), (4, 0) and
[2011 (T-II)]
(–6, 10) are collinear. [2011 (T-II)]
33. One end of a diameter of a circle is at (2, 3) and
15. Find the ratio in which the y-axis divides the join the centre is (–2, 5). What are the cooridnates of
of (5, – 6) and (–1, – 4). [2011 (T-II)] the other end of the diameter? [2011 (T-II)]
16. Find the co-ordinates of a point A, where AB is 34. A point P is at a distance of 10 from the point
the diameter of a circle whose centre is O(2, –3) (2, 3). Find the coordinates of the point P if its y
and B is (1, 4). [2011 (T-II)] coordinate is twice its x coordinate.
17. Using section formula, show that the points [2011 (T-II)]
A(–3, 1), B (1, 3) and C(–1, 1) are collinear. 35. Find the coordinates of the point B, if the point
6
 P(– 4, 1) divides the line segment joining the 39. Find the points on the x-axis which are at a distnace
points A(2, –2) and B in the ratio 3 : 5. of 2 5 from the point (7, –4). How many such
[2011 (T-II)] points are there? [2011 (T-II)]
36. Find the third vertex of the triangle ABC if two 40. If A and B are the points (–2, –2) and (2, –4)
of its vertices are at A(–3, 1) and B (0, 2) and respectively, find the coordinates of P on the line
3
3 1 segment AB such that AP = AB.
the mid-point of BC is at D  , −  . 7
2 2 [2011 (T-II)]
[2011 (T-II)]
41. Find a point on x-axis which is equidistant from
37. Find the value of s if the point P(0, 2) is equidistant (–3, –4) and (4, –3). [2011 (T-II)]
from Q (3, s) and R(s, 5).
42. Find the value of a so that the point (3, a) lies
38. Find the perimeter of the triangle formed by the on the line represented by 2x – 3y = 5. [2009]
points (0, 0), (1, 0), (0, 1). [2011 (T-II)]

N
Short Answer Type Questions [3 Marks]

A
SH
A. Important Questions

A
1. If P(x, y) is the mid-point of the line segment 10. If the mid-points of the sides of a triangle are
joining the points A(3, 4) and B(k, 6) and

K
3   11 
x + y = 10, then find the value of k. (2, 3),  ,4  and  , 5  , find the centroid
2 

A
2 
2. Find the coordinates of the centre of a circle of the triangle.
passing through the points A(2, 1), B(5, –8) and PR
11. Using coordinate geometry, prove that the
C(2, –9). Also, find the radius of the circle.
diagonals of a rectangle bisect each other and are
S
3. Find the points on the x-axis which are at a distance equal.
ER

of units from the point (7, –4). How many 12. Find the coordinates of the points C on the line
such points are there ? segment joining the points A(–1, 3) and B(2, 5)
4. Find the area of the triangle ABC with A(1, – 4) 3
TH

and the mid-points of the sides through A being such that AC = AB.
5
(2, –1) and (0, –1).
13. Show that (4, –1), (6, 0), (7, 2) and (5, 1) are the
O

5. Find the coordinates of the points of trisection of

vertices of a rhombus. Is it a square?
R

the line segment joining (4, –1) and (– 2, –3).

B

6. If three points (x1, y1), (x2, y2) and (x3, y3) lie on 14. The coordinates of the three vertices A, B and C
the same line, prove that [HOTS] of a parallelogram ABCD are (x1, y1), (x2, y2) and
L

y 2 − y3 y3 − y1 y1 − y2 (x3, y3) respectively. Find the coordinates of the

YA

+ + = 0
x2 x3 x3 x1 x1 x2 fourth vertex D in terms of x1, x2, x3, y1, y2, y3.
[HOTS]
O

7. The centre of a circle is (2a, a – 7). Find the

 1 5 7 7
G

value of a, if the circle passes through the point

15. If D  − ,  , E(7, 3) and F  2 ,  are the
2
(11, –9) and has diameter units. 2 2

8. What type of triangle is formed by the points mid-points of sides of DABC, find the area of

A ( 2, 2 ) B( 2, − 2 ) and C (− )
6, 6 . DABC.

16. If the line segment joining the points

3 5  A(3a + 1, –3) and B(8a, 5) is divided by the
9. Find the ratio in which the point M  , 
4 12  point P(9a – 2, –b) in the ratio 3 : 1, find the
divides the line segment joining the points A values of a and b.
1 3
 ,  and B(2, –5).
2 2

7
 B. Questions From CBSE Examination Papers

1. Find the value(s) of x for which distance between A(–1, –1), B(–1, 4), C(5, 4) and D(5, –1). P, Q, R
the points P(2, –3) and Q(x, 5) is 10 units. and S are the mid-points of AB, BC, CD and DA
[2011 (T-II)] respectively. Is the quadrilateral PQRS a square,
2. Find a relation between x and y such that the a rectangle or a rhombus? Justify your answer.
point P(x, y) is equidistant from the points [2011 (T-II)]
A(2, 5) and B(–3, 7). [2011 (T-II)] 18. The line segment joining the points A(2, 1)
3. Find the area of a triangle ABC whose vertices and B(5, –8) is trisected at the points P and Q,
are A(1, –1), B (–4, 6) and C(–3, –5). where P is nearer to A. If point P lies on the line
2x – y + k = 0, find the value of k.
4. If the points (1, 4), (r, –2) and (–3, 16) are
collinear, find the value of ‘r’. [2011 (T-II)] [2011 (T-II)]
5. If A (3, 0), B (4, 5), C (–1, 4) and D (–2, –1) 19. If P(x, y) is any point on the line segment joining
be four points in a plane, show that ABCD is a the points A(a, 0) and B(0, b), then show that

N
rhombus but not a square. [2011 (T-II)]
x y

A
6. The mid-point of the line segment joining points + =1
A(x, y + 1) and B(x + 1, y + 2) is C. Find a b [2011 (T-II)]

SH

the value of x and y if the coordinates of C are
(3/2, 5/2). [2011 (T-II)] 20. If C is a point lying on the line segment AB
joining A(1, 1), B(2, 3) such that 3AC = BC,

A
7. Find the area of rhombus ABCD if its vertices then find co-ordinates of C. [2011 (T-II)]

K
are A(3, 0), B(4, 5), C(–1, 4), D(–2, –1).
[2011 (T-II)] 21. In the figure, in DABC, D and E are the mid-

A
8. Find the area of the triangle formed by joining points of the sides BC and AC respectively. Find
the mid-points of the sides of the triangle
PR the length of DE. Prove that DE =
1
AB.
whose vertices are (0, –1), (2, 1) and (0, 3). 2
[2011 (T-II)] [2011 (T-II)]
S

9. The area of a triangle whose vertices are

ER

(–2, –2), (–1, –3) and (x, 0) is 3 square units.

Find the value of x. [2011 (T-II)]
TH

10. If P(2, 1), Q(4, 2), R(5, 4) and S(3, 3) are vertices
of a quadrilateral PQRS, find the area of the
quadrilateral PQRS. [2011 (T-II)]
O

11. If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices
R

of a parallelogram taken in order, find x and y.

B

12. Show that the points P(0, –2), Q(3,1), R(0, 4) and 22. Show that the points (–4, 0), (4, 0) and (0, 3) are
S(–3, 1) are the vertices of a square. PQRS. vertices of an isosceles triangle. [2011 (T-II)]
L

[2011 (T-II)]
YA

23. Find the value of p so that the points with

13. The points A (2, 9), B (a, 5) and C (5, 5) are
the vertices of a triangle ABC, right angled at B. coordinates (3, 5), (p, 6) and  1 , 15  are
 
O

Find the value of a and hence the area of triangle 2 2

ABC. [2011 (T-II)]
G

collinear. [2011 (T-II)]

14. Find the value of k for which the points 24. The base BC of an equilateral triangle ABC lies
A(–1, 3), B(2, k) and C(5, –1) are collinear. on y-axis. The co-ordinates of the point C are
[2011 (T-II)] (0, –3). If origin is the mid-point of BC, find the
15. Find the lengths of the medians AD and BE of coordinates of points A and B. [2011 (T-II)]
the triangle ABC whose vertices are A(1, –1), 25. Determine the ratio in which the point (x, 2)
B(0, 4) and C(–5, 3). [2011 (T-II)] divides the line segment joining the points
16. The mid-points of the sides AB, BC and CA of (–3, –4) and (3, 5). Also, find x. [2011 (T-II)]
a triangle ABC are D(2, 1), E(1, 0) and F(–1, 3)
respectively. Find the coordinates of the vertices 26. Show that the points A(a, a), B (–a, –a) and C
of the triangle ABC. [2011 (T-II)] ( )
− a 3 , a 3 form an equilateral triangle.
17. ABCD is a rectangle formed by joining the points [2011 (T-II)]
8
 27. Point P divides the line segment joining the points 37. The point R divides the line segment AB where
AP 1 3
A(2, 1) and B(5, –8) such that = . If P lies A(– 4, 0), B(0, 6) are such that AR = AB.
PB 3 4
Find the coordinates of R. [2008]
on the line 2x – y + k = 0, find the value of k.
[2011 (T-II)] 38. The coordinates of A and B are (1, 2) and (2, 3)
28. If the vertices of a triangle are (2, 4) (5, k), respectively. If P lies on AB, find the coordinates
(3, 10) and its area is 15 square units, find value AP 4
of k. [2011 (T-II)] of P such that = . [2008]
PB 3
29. Find the coordinates of centre of the circle passing 39. If A(4, –8), B(3, 6) and C(5, – 4) are the vertices
through the point (0, 0), (–2, 1) and (–3, 2). Also of DABC, D is the mid-point of BC and P is a
find its radius. [2011 (T-II)] AP
point on AD joined such that = 2. Find the
30. If mid-points of sides DPQR are (1, 2), (0, 1), PD
(1, 0), then find the coordintes of the three vertices coordinates of P. [2008]

N
of triangle PQR. [2011 (T-II)] 40. The line segment joining the points A(2, 1) and

A
31. Prove that the points (0, 0), (5, 5), (–5, 5) are the B(5, –8) is trisected at the points P and Q such
vertices of a right angled triangle. [2011 (T-II)] that P is nearer to A. If P also lies on the line

SH
given by 2x – y + k = 0, find the value of k.
32. Find the co-ordinates of the points which divide [2009]
the line segment joining A(–2, 2) and B(2, 8) into

A
four equal parts. [2011 (T-II)] 41. If P(x, y) is any point on the line joining the points

K
x y
33. If A (4, –6), B (3, –2) and C(5, 2) are the vertices A(a, 0) and B(0, b), then show that + =1.

A
of a DABC, then verify the fact that a median a b
of DABC divides it into two triangles of equal PR
[2009]
areas. [2011 (T-II)] 42. If the points (p, q), (m, n) and (p – m, q – n) are
34. If the points (10, 5), (8, 4) and (6, 6) are the collinear, show that pn = qm. [2010]
S
mid-points of the sides of a triangle, find its
vertices. [2006] 43. Point P divides the line segment joining the points
ER

35. In what ratio does the line x – y – 2 = 0 divide AP k

A(–1, 3) and B(9, 8) such that = . If P lies
the line segment joining (3, –1) and (8, 9) ? PB 1
TH

[2007] on the line x – y + 2 = 0, find the value of k.

36. Three consecutive vertices of a parallelogram are [2010]
O

(– 2, 1); (1, 0) and (4, 3). Find the coordinate of

the fourth vertex. [2007]
R
B

Long Answer Type Questions [4 Marks]

L
YA

A. Important Questions
O

1. The points A(x1, y1), B(x2, y2) and C(x3, y3) are 2. If P and Q are two points whose coordinates are
the vertices of DABC.  a 2a 
G

(at2, 2at) and  2 , −  respectively and S is the

(i) The median from A meets BC at D. What t t 1 1
are the coordinates of the point D? point (a, 0), show that + is independent
(ii) Find the coordinates of the point P on AD of t. SP SQ [HOTS]
such that AP : PD = 2 : 1. 3. If (– 4, 3) and (4, 3) are two vertices of an
(iii) Find the coordinates of points Q and R on equilateral triangle, find the coordinates of the third
medians BE and CF respectively such that vertex given that the origin lies in the interior of
BQ : QE = 2 : 1 and CR : RF = 2 : 1. the triangle.
(iv) What are the coordinates of the centroid of 4. If the points (x, y), (x1, y1) and (x – x1, y – y1)
the triangle ABC? [HOTS] are collinear, show that xy1 = x1y. Also, show that
the line joining the given points passes through
the origin. [HOTS]

9
 5. Mr. Aggarwal starts walking from his home to office at (13, 26) and coordinates are in km.
office. Instead of going to the office directly, he [HOTS]
goes to a bank first, from there to his daughter’s 6. Find the centre of a circle passing through the
school and then reaches the office. What is the points (6, –6), (3, –7) and (3, 3).
extra distance travelled by Mr. Aggarwal in
reaching his office ? Assume that all distances 7. If the coordinates of the mid-points of the sides
covered are in straight lines. If the house is situated of a triangle are (1, 1), (2, –3) and (3, 4), find its
at (2, 4), bank at (5, 8), school at (13, 14) and centroid.

Formative Assessment

Activity-1
Coordinates of
Objective : To verify the distance formula and section

N
formula. Or to verify the following : P (–1, 9/2)

A
(i) The distance between the points (x1, y1) and
(x2, y2) is Q (3, 9/2)

SH
( x2 − x1 )2 + ( y2 − y1 )2 R (–1, 1)

A
(ii) The coordinates of the point P, which di-
S (3, 1)

K
vides the line segment joining the points

A
A(x1, y1) and B(x2, y2) in the ratio m : n are Observations :
 mx2 + nx1 my2 + ny1  PR 1. Using a ruler, measure the length of AB, BC
 m+n , m+n . and CA
 
(iii) The coordinates of the mid-points P of the We have AB = BC = CA = 8 cm.
S
line segment joining the points A (x1, y1) and 2. Now, using formula
x +x y + y2 
ER

B (x2, y2) are  1 2 , 1 .

2  (5 + 3)2 + (1 − 1)
2
 2 AB = cm = 8 cm
Materials Required : Graph paper/squared paper,
TH

geometry box, etc. BC = (1 − 5)2 + (8 − 1) cm

2

Procedure : = 16 + 49 cm = 65 cm = 8.06 cm
O

1. Take a 1 cm squared paper and on it draw the

R

coordinate axes XOX’ and YOY’. = 8 cm

B

2. Plot the points A (–3, 1), B (5, 1) and C (1, 8)

on the squared paper and join AB, BC and AC
CA = (1 + 3)2 + (8 − 1) cm
2
L

to get a ∆ABC. = 16 + 49 cm = 65 cm = 8.06 cm = 8 cm

YA

3. Now using a pair of compasses, find the mid- We see that in both the cases, the length of
points of AC and BC. Mark these mid-points as each side comes out to be 8 cm.
O

P and Q respectively. 3. Since P and Q are mid-points of AC and BC

G

1
4. Mark the point R on AB such that AR = AB respectively, therefore, using formula, the coor-
4
or R divides AB in the ratio 1 : 3. Similarly,  1 − 3 8 + 1  9
dinates of P are 
 2
,  or  −1,  the
2 2
3
mark S on AB such that AS = AB or S divides 1+ 5 8 +1  9
AB in the ratio 3 : 1. 4 coordinate of Q are  ,  or  3, 
 2 2   2
5. From the graph paper, write the coordinates of Also, from the table : the coordinates of P and
P, Q, R and S. Q are same as obtained above.
4. Since R divides AB in the ratio 1 : 3, therefore,

10
  1 × 5 + 3 × ( −3) 1 × 1 + 3 × 1
coordinates of R (using formula) are  , or (–1, 1).
 1+ 3 1 + 3 
Similarly, S divides AB in the ratio 3 : 1, therefore, coordinates of S (using formula) are:
 3 × 5 + 1 × ( −3) 3 × 1 + 1 × 1
 , or (3, 1).
 3 +1 3 + 1 
Also, from the table, the coordinates of R and S are same as obtained above.

N
A
SH
A
K
A
PR
S
ER
TH
O
R
B
L
YA
O

Conclusion : From the above activity it is verified segment joining the points (x1, y1) and (x2, y2)
G

that :  x1 + x2 y1 + y2 
are  , .
(i) the distance of the line segments joining  2 2 
the points (x1, y1) and (x2, y2) is given by Do Yourself : Draw three different triangles on a
( x2 − x1 ) + ( y2 − y1 )
2 2
paper and in each case verify the above formulae.

(ii) If a point P divides the line segment join- Activity-2

ing the points (x1, y1) and (x2, y2) in the
ratio m : n, then the coordinates of P are Objective : To verify the following formula :
 mx2 + nx1 my2 + ny1  The area of a triangle having ver-
 m+n , m + n 
. tices (x 1 , y 1 ), (x 2 , y 2 ) and (x 3 , y 3 ) is given by

1
(iii) The coordinates of the mid-point of the line [x (y – y3) + x2 (y3 – y1) + x3 (y1 – y2)]
2 1 2
11
Materials Required : Graph paper/squared paper,
geometry box etc. = (36 − 8)(8 − 1) cm 2

Procedure : 1. Take a 1 cm squared paper and on it = 28 × 7 cm 2 = 14 cm 2

draw the coordinate axes XOX’ and YOY’. 4. Now, find the area of the triangle ABC, using
Case I. the formula
1. Plot the points A (–3, 0), B (4, 0) and C(0, 4) 1
on the squared paper and join AB, BC and AC [x (y – y3) – x2(y3 – y1) + x3(y1 – y2)]
2 1 2
to get a scalene triangle ABC. 1
Area of ∆ABC = [–3(0 – 4) + 4(4 – 0)
2
2
+ 0(0 – 0)] cm
1
= [12 + 16] cm2 = 14 cm2.
2
Observations : From (3) and (4), we see that the area
of ∆ABC comes out to be same on both the cases.
Case II.

N
1. Plot the points P (–2, 3), Q (–2, 0) and R

A
(2, 0) and join PQ, QR and PR to get the right
triangle PQR.

SH
A
K
A
2. Find AB, BC and AC using the distance for-
mula. PR
AB = ( − 3 − 4 )2 + ( 0 − 0) cm
2
S
= 49 cm = 7 cm
ER

BC = ( 4 − 0 )2 + ( 0 − 4) cm
2
TH

= 16 + 16 cm = 4 2 cm
2. Find PQ, QR and PR using the distance for-
AC = ( − 3 − 0) 2
+ ( 0 − 4) cm
2 mula.
O

PQ = ( − 2 + 2 )2 + (3 − 0) cm
2
R

= 9 + 16 cm = 5 cm
= 9 cm = 3 cm
B

3. Find the area of ∆ABC using Heron’s for-

mula. QR = ( − 2 − 2 )2 + ( 0 − 0) cm
2
L

Here, a = 7, b = 4 2 , c = 5
YA

= 16 cm = 4 cm
7+4 2 +5
∴ s = =6+2 2 PR = ( − 2 − 2 )2 + (3 − 0) cm
2
O

= 16 + 9 cm = 5 cm
G

∴ Area of ∆ABC
3. Find the area of ∆PQR using Heros’s
= s ( s − a ) ( s − b )( s − c ) formula:
Here, a = 3 cm, b = 4 cm, c = 5 cm
= (6 + 2 2 ) (6 + 2 2 −7 ) (6 + 2 2−4 2 ) 3+4+5
(6 + 2 2 − 5 ) cm2
∴ s =
2
cm = 6 cm

∴ Area of ∆PQR = s ( s − a )( s − b )( s − c )
= (6 + 2 2 ) ( 2 2 − 1) (6 − 2 2 ) ( 2 2 + 1) cm2
6 ( 6 − 3)( 6 − 4)( 6 − 5) cm2
=
= (6 + 2 2 ) (6 − 2 2 ) ( 2 2 −1 ) (2 )
2 + 1 cm2 = 6 × 3 × 2 × 1 cm2 = 6 cm2

12
 4. Now, find the area of ∆PQR using the formula For each rectangle, draw a diagonal and count
1 the number of squares through which the diagonal
[x (y – y3) + x2(y3 – y1) + x3(y1 – y2)] passes.
2 1 2
1 The dimensions of a rectangle is 120 × 91. How
∴ Area of ∆PQR = [–2 (0 – 0) – 2 (0 – 3)
2 many squares will the diagonal pass through?
1
+ 2 (3 – 0)] cm2 = [6 + 6] = 6 cm2 If the dimensions of a rectangle are m × n, where
2
Observations :From (3) and (4) above, we see that m and n are co-primes, then how many squares will the
area of ∆PQR comes out to be same in both the diagonal pass through?
cases.
Conclusion : F r o m t h e a b o v e a c t i v i t y i t i s
verified that the area of a triangle having vertices
(x1, y1), (x2, y2) and (x3, y3) is given by
1
[x (y – y3) + x2(y3 – y1) + x3(y1 – y2)]

N
2 1 2

A
Investigation

SH
On a squared / graph paper, draw a rectangle of
dimensions 4 × 3. Draw one of the diagonals of the

A
rectangle. The diagonal passes through 6 squares.

K
Now on a squared paper draw several rectangles Puzzle
of different sizes. The length (l) and breadth (b) of each

A
rectangle must be a whole number of squares with a A merchant has nine gold coins which look
common factor of 1 only. For example, sides 4 squares
by 9 squares is acceptable but 4 squares by 6 squares
PR
identical but in fact one of the coins is an underweight
fake. Investigate how the merchant can use only a
is not because 4 and 6 have a common factor 2. balance to find the fake coin in just two weighings.
S
ER
TH
O
R
B
L
YA
O
G

13
Class X Chapter 7 – Coordinate Geometry Maths

Exercise 7.1
Question 1:
Find the distance between the following pairs of points:
(i) (2, 3), (4, 1) (ii) (−5, 7), (−1, 3) (iii) (a, b), (− a, − b)
Answer:
(i) Distance between the two points is given by

(ii) Distance between is given by

(iii) Distance between is given by

Question 2:
Find the distance between the points (0, 0) and (36, 15). Can you now find the
distance between the two towns A and B discussed in Section 7.2.
Answer:

Distance between points

Page 1 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Yes, we can find the distance between the given towns A and B.
Assume town A at origin point (0, 0).
Therefore, town B will be at point (36, 15) with respect to town A.
And hence, as calculated above, the distance between town A and B will be
39 km.
Question 3:
Determine if the points (1, 5), (2, 3) and (− 2, − 11) are collinear.
Answer:
Let the points (1, 5), (2, 3), and (−2, −11) be representing the vertices A, B, and C
of the given triangle respectively.

Let

Therefore, the points (1, 5), (2, 3), and (−2, −11) are not collinear.
Question 4:
Check whether (5, − 2), (6, 4) and (7, − 2) are the vertices of an isosceles triangle.
Answer:
Let the points (5, −2), (6, 4), and (7, −2) are representing the vertices A, B, and C
of the given triangle respectively.

Page 2 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

As two sides are equal in length, therefore, ABCis an isosceles triangle.

Question 5:
In a classroom, 4 friends are seated at the points A, B, C and D as shown in the
following figure. Champa and Chameli walk into the class and after observing for a
few minutes Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli
disagrees.
Using distance formula, find which of them is correct.

Answer:
It can be observed that A (3, 4), B (6, 7), C (9, 4), and D (6, 1) are the positions of
these 4 friends.

Page 3 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

It can be observed that all sides of this quadrilateral ABCD are of the same length
and also the diagonals are of the same length.
Therefore, ABCD is a square and hence, Champa was correct

Question 6:
Name the type of quadrilateral formed, if any, by the following points, and give
reasons for your answer:
(i) (− 1, − 2), (1, 0), (− 1, 2), (− 3, 0)
(ii) (− 3, 5), (3, 1), (0, 3), (− 1, − 4)
(iii) (4, 5), (7, 6), (4, 3), (1, 2)

Page 4 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Answer:
(i) Let the points (−1, −2), (1, 0), (−1, 2), and (−3, 0) be representing the vertices
A, B, C, and D of the given quadrilateral respectively.

It can be observed that all sides of this quadrilateral are of the same length and also,
the diagonals are of the same length. Therefore, the given points are the vertices of
a square.
(ii)Let the points (− 3, 5), (3, 1), (0, 3), and (−1, −4) be representing the vertices
A, B, C, and D of the given quadrilateral respectively.

It can be observed that all sides of this quadrilateral are of different lengths.
Therefore, it can be said that it is only a general quadrilateral, and not specific such
as square, rectangle, etc.
(iii)Let the points (4, 5), (7, 6), (4, 3), and (1, 2) be representing the vertices A, B,
C, and D of the given quadrilateral respectively.

Page 5 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

It can be observed that opposite sides of this quadrilateral are of the same length.
However, the diagonals are of different lengths. Therefore, the given points are the
vertices of a parallelogram.

Question 7:
Find the point on the x-axis which is equidistant from (2, − 5) and (− 2, 9).
Answer:
We have to find a point on x-axis. Therefore, its y-coordinate will be 0.

Let the point on x-axis be .

By the given condition, these distances are equal in measure.

Page 6 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Therefore, the point is (− 7, 0).

Question 8:
Find the values of y for which the distance between the points P (2, − 3) and Q (10,
y) is 10 units.
Answer:
It is given that the distance between (2, −3) and (10, y) is 10.

Question 9:
If Q (0, 1) is equidistant from P (5, − 3) and R (x, 6), find the values of x. Also find
the distance QR and PR.
Answer:

Therefore, point R is (4, 6) or (−4, 6).

Page 7 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

When point R is (4, 6),

When point R is (−4, 6),

Question 10:
Find a relation between x and y such that the point (x, y) is equidistant from the
point (3, 6) and (− 3, 4).
Answer:
Point (x, y) is equidistant from (3, 6) and (−3, 4).

Page 8 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Exercise 7.2
Question 1:
Find the coordinates of the point which divides the join of (− 1, 7) and (4, − 3) in
the ratio 2:3.
Answer:
Let P(x, y) be the required point. Using the section formula, we obtain

Therefore, the point is (1, 3).

Question 2:
Find the coordinates of the points of trisection of the line segment joining (4, − 1)
and (− 2, − 3).
Answer:

Let P (x1, y1) and Q (x2, y2) are the points of trisection of the line segment joining
the given points i.e., AP = PQ = QB
Therefore, point P divides AB internally in the ratio 1:2.

Point Q divides AB internally in the ratio 2:1.

Page 9 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Question 3:
To conduct Sports Day activities, in your rectangular shaped school ground ABCD,
lines have been drawn with chalk powder at a distance of 1 m each. 100 flower pots
have been placed at a distance of 1 m from each other along AD, as shown in the

following figure. Niharika runs the distance AD on the 2nd line and posts a green

flag. Preet runs the distance AD on the eighth line and posts a red flag. What is
the distance between both the flags? If Rashmi has to post a blue flag exactly
halfway between the line segment joining the two flags, where should she post her
flag?

Page 10 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Answer:

It can be observed that Niharika posted the green flag at of the distance AD i.e.,

m from the starting point of 2nd line. Therefore, the coordinates of

this point G is (2, 25).

Similarly, Preet posted red flag at of the distance AD i.e., m from

th
the starting point of 8 line. Therefore, the coordinates of this point R are (8, 20).
Distance between these flags by using distance formula = GR

=
The point at which Rashmi should post her blue flag is the mid-point of the line
joining these points. Let this point be A (x, y).

Therefore, Rashmi should post her blue flag at 22.5m on 5th line
Question 4:
Find the ratio in which the line segment joining the points (− 3, 10) and (6, − 8) is
divided by (− 1, 6).
Answer:
Let the ratio in which the line segment joining (−3, 10) and (6, −8) is divided by
point (−1, 6) be k : 1.

Page 11 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Question 5:
Find the ratio in which the line segment joining A (1, − 5) and B (− 4, 5) is divided
by the x-axis. Also find the coordinates of the point of division.
Answer:
Let the ratio in which the line segment joining A (1, −5) and B (−4, 5) is divided by

x-axisbe .

Therefore, the coordinates of the point of division is .

We know that y-coordinate of any point on x-axis is 0.

Therefore, x-axis divides it in the ratio 1:1.

Division point =

Question 6:
If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order,
find x and y.
Answer:

Page 12 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Let (1, 2), (4, y), (x, 6), and (3, 5) are the coordinates of A, B, C, D vertices of a
parallelogram ABCD. Intersection point O of diagonal AC and BD also divides these
diagonals.
Therefore, O is the mid-point of AC and BD.
If O is the mid-point of AC, then the coordinates of O are

If O is the mid-point of BD, then the coordinates of O are

Since both the coordinates are of the same point O,

Question 7:
Find the coordinates of a point A, where AB is the diameter of circle whose centre is
(2, − 3) and B is (1, 4)
Answer:
Let the coordinates of point A be (x, y).
Mid-point of AB is (2, −3), which is the center of the circle.

Page 13 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Question 8:
If A and B are (− 2, − 2) and (2, − 4), respectively, find the coordinates of P such

that and P lies on the line segment AB.

Answer:

The coordinates of point A and B are (−2, −2) and (2, −4) respectively.

Since ,
Therefore, AP: PB = 3:4
Point P divides the line segment AB in the ratio 3:4.

Page 14 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Question 9:
Find the coordinates of the points which divide the line segment joining A (− 2, 2)
and B (2, 8) into four equal parts.
Answer:

From the figure, it can be observed that points P, Q, R are dividing the line segment
in a ratio 1:3, 1:1, 3:1 respectively.

Question 10:
Find the area of a rhombus if its vertices are (3, 0), (4, 5), (− 1, 4) and (− 2, −1)

taken in order. [Hint: Area of a rhombus = (product of its diagonals)]

Page 15 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Answer:

Let (3, 0), (4, 5), (−1, 4) and (−2, −1) are the vertices A, B, C, D of a rhombus
ABCD.

Page 16 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Exercise 7.3
Question 1:
Find the area of the triangle whose vertices are:
(i) (2, 3), (− 1, 0), (2, − 4) (ii) (− 5, − 1), (3, − 5), (5, 2)
Answer:
(i) Area of a triangle is given by

(ii)

Question 2:
In each of the following find the value of ‘k’, for which the points are collinear.
(i) (7, − 2), (5, 1), (3, − k) (ii) (8, 1), (k, − 4), (2, − 5)
Answer:
(i) For collinear points, area of triangle formed by them is zero.
Therefore, for points (7, −2) (5, 1), and (3, k), area = 0

Page 17 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

(ii) For collinear points, area of triangle formed by them is zero.

Therefore, for points (8, 1), (k, −4), and (2, −5), area = 0

Question 3:
Find the area of the triangle formed by joining the mid-points of the sides of the
triangle whose vertices are (0, − 1), (2, 1) and (0, 3). Find the ratio of this area to
the area of the given triangle.
Answer:

Let the vertices of the triangle be A (0, −1), B (2, 1), C (0, 3).
Let D, E, F be the mid-points of the sides of this triangle. Coordinates of D, E, and F
are given by

Page 18 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Question 4:
Find the area of the quadrilateral whose vertices, taken in order, are (− 4, − 2), (−
3, − 5), (3, − 2) and (2, 3)
Answer:

Let the vertices of the quadrilateral be A (−4, −2), B (−3, −5), C (3, −2), and D (2,
3). Join AC to form two triangles ∆ABC and ∆ACD.

Page 19 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Question 5:
You have studied in Class IX that a median of a triangle divides it into two triangles
of equal areas. Verify this result for ∆ABC whose vertices are A (4, − 6), B (3, − 2)
and C (5, 2)
Answer:

Let the vertices of the triangle be A (4, −6), B (3, −2), and C (5, 2).
Let D be the mid-point of side BC of ∆ABC. Therefore, AD is the median in ∆ABC.

Page 20 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

However, area cannot be negative. Therefore, area of ∆ABD is 3 square units.

However, area cannot be negative. Therefore, area of ∆ADC is 3 square units.

Clearly, median AD has divided ∆ABC in two triangles of equal areas.

Page 21 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Exercise 7.4
Question 1:
Determine the ratio in which the line 2x + y − 4 = 0 divides the line segment joining
the points A(2, − 2) and B(3, 7)
Answer:
Let the given line divide the line segment joining the points A(2, −2) and B(3, 7) in a
ratio k : 1.

Coordinates of the point of division

This point also lies on 2x + y − 4 = 0

Therefore, the ratio in which the line 2x + y − 4 = 0 divides the line segment joining
the points A(2, −2) and B(3, 7) is 2:9.
Question 2:
Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are collinear.
Answer:
If the given points are collinear, then the area of triangle formed by these points will
be 0.

Page 22 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

This is the required relation between x and y.

Question 3:
Find the centre of a circle passing through the points (6, − 6), (3, − 7) and (3, 3).
Answer:
Let O (x, y) be the centre of the circle. And let the points (6, −6), (3, −7), and (3, 3)
be representing the points A, B, and C on the circumference of the circle.

Page 23 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

On adding equation (1) and (2), we obtain

10y = −20
y = −2
From equation (1), we obtain
3x − 2 = 7
3x = 9
x=3
Therefore, the centre of the circle is (3, −2).

Question 4:
The two opposite vertices of a square are (− 1, 2) and (3, 2). Find the coordinates of
the other two vertices.
Answer:

Let ABCD be a square having (−1, 2) and (3, 2) as vertices A and C respectively. Let
(x, y), (x1, y1) be the coordinate of vertex B and D respectively.
We know that the sides of a square are equal to each other.
∴ AB = BC

We know that in a square, all interior angles are of 90°.

Page 24 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

In ∆ABC,
AB2 + BC2 = AC2

⇒ 4 + y2 + 4 − 4y + 4 + y2 − 4y + 4 =16
⇒ 2y2 + 16 − 8 y =16
⇒ 2y2 − 8 y = 0
⇒ y (y − 4) = 0
⇒ y = 0 or 4
We know that in a square, the diagonals are of equal length and bisect each other at
90°. Let O be the mid-point of AC. Therefore, it will also be the mid-point of BD.

⇒ y + y1 = 4
If y = 0,
y1 = 4
If y = 4,
y1 = 0
Therefore, the required coordinates are (1, 0) and (1, 4).
Question 5:
The class X students of a secondary school in Krishinagar have been allotted a
rectangular plot of land for their gardening activity. Saplings of Gulmohar are planted

Page 25 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

on the boundary at a distance of 1 m from each other. There is a triangular grassy

lawn in the plot as shown in the following figure. The students are to sow seeds of
flowering plants on the remaining area of the plot.

(i) Taking A as origin, find the coordinates of the vertices of the triangle.
(ii) What will be the coordinates of the vertices of ∆ PQR if C is the origin?
Also calculate the areas of the triangles in these cases. What do you observe?
Answer:
(i) Taking A as origin, we will take AD as x-axis and AB as y-axis. It can be observed
that the coordinates of point P, Q, and R are (4, 6), (3, 2), and (6, 5) respectively.

(ii) Taking C as origin, CB as x-axis, and CD as y-axis, the coordinates of vertices P,

Q, and R are (12, 2), (13, 6), and (10, 3) respectively.

Page 26 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

It can be observed that the area of the triangle is same in both the cases.

Question 6:
The vertices of a ∆ABC are A (4, 6), B (1, 5) and C (7, 2). A line is drawn to intersect

sides AB and AC at D and E respectively, such that . Calculate the area

of the ∆ADE and compare it with the area of ∆ABC. (Recall Converse of basic
proportionality theorem and Theorem 6.6 related to
ratio of areas of two similar triangles)
Answer:

Given that,

Page 27 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Therefore, D and E are two points on side AB and AC respectively such that they
divide side AB and AC in a ratio of 1:3.

Clearly, the ratio between the areas of ∆ADE and ∆ABC is 1:16.
Alternatively,
We know that if a line segment in a triangle divides its two sides in the same ratio,
then the line segment is parallel to the third side of the triangle. These two triangles
so formed (here ∆ADE and ∆ABC) will be similar to each other.

Page 28 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Hence, the ratio between the areas of these two triangles will be the square of the
ratio between the sides of these two triangles.

Therefore, ratio between the areas of ∆ADE and ∆ABC =

Question 7:
Let A (4, 2), B (6, 5) and C (1, 4) be the vertices of ∆ABC.
(i) The median from A meets BC at D. Find the coordinates of point D.
(ii) Find the coordinates of the point P on AD such that AP: PD = 2:1
(iii) Find the coordinates of point Q and R on medians BE and CF respectively such
that BQ: QE = 2:1 and CR: RF = 2:1.
(iv) What do you observe?
(v) If A(x1, y1), B(x2, y2), and C(x3, y3) are the vertices of ∆ABC, find the coordinates
of the centroid of the triangle.
Answer:

(i) Median AD of the triangle will divide the side BC in two equal parts.
Therefore, D is the mid-point of side BC.

(ii) Point P divides the side AD in a ratio 2:1.

Page 29 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

(iii) Median BE of the triangle will divide the side AC in two equal parts.
Therefore, E is the mid-point of side AC.

Point Q divides the side BE in a ratio 2:1.

Median CF of the triangle will divide the side AB in two equal parts. Therefore, F is
the mid-point of side AB.

Point R divides the side CF in a ratio 2:1.

(iv) It can be observed that the coordinates of point P, Q, R are the same.
Therefore, all these are representing the same point on the plane i.e., the centroid of
the triangle.
(v) Consider a triangle, ∆ABC, having its vertices as A(x1, y1), B(x2, y2), and C(x3,
y3).
Median AD of the triangle will divide the side BC in two equal parts. Therefore, D is
the mid-point of side BC.

Page 30 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

Let the centroid of this triangle be O.

Point O divides the side AD in a ratio 2:1.

Question 8:
ABCD is a rectangle formed by the points A (− 1, − 1), B (− 1, 4), C (5, 4) and D (5,
− 1). P, Q, R and S are the mid-points of AB, BC, CD, and DA respectively. Is the
quadrilateral PQRS is a square? a rectangle? or a rhombus? Justify your answer.
Answer:

Page 31 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)
Class X Chapter 7 – Coordinate Geometry Maths

It can be observed that all sides of the given quadrilateral are of the same measure.
However, the diagonals are of different lengths. Therefore, PQRS is a rhombus.

Page 32 of 32
Website: www.vidhyarjan.com Email: contact@vidhyarjan.com Mobile: 9999 249717

Head Office: 1/3-H-A-2, Street # 6, East Azad Nagar, Delhi-110051

(One Km from ‘Welcome Metro Station)