Chapter 7: Coordinates Geometry – Chapter-Wise Important Questions and Answers | Class 10 Mathematics CBSE Solutions
CBSE Class 10 Mathematics Chapter 7: Coordinates Geometry Important Questions with Solutions and Answers
Chapter 7: COORDINATES GEOMETRY
SECTION = A
Question: The point 
is equidistant from the points 
and 
,then : [ SEBA 2020]
(a) 
(b) 
(c) 
(d) 
Answer: (c) .
[ Let the distance of 
from 
and 
are equal .
A/Q , 
]
Question: The distance between the points 
and 
is : [SEBA 2019]
(a) 10 units (b) 8 units (c) 6 units (d) 2 units
Answer: (b) 10
[ The distance between the points 
units ]
Question: If 
is mid-point of the line segment joining the points 
and 
,then 
is :
(a) – 1 (b) 1 (c) 2 (d) – 2
Answer: (b) 1
[ We have , 
]
Question: The distance between the points 
and 
is : [SEBA 2015 , 18]
(a) 2 (b) 
(c) 1 (d) 0
Answer: (b)
[ The distance between the points 
unit ]
Question: The line segment joining the points 
and 
in the ratio 2 : 3 , then the coordinate of the point is :
(a) 
(b) 
(c) 
(d) 
Answer: (d) .
[ Here, 
, 
, 
, 
, 
and 
and 
Question: The distance between the point 
and 
is :
(a) 
(b) 
(c) 
(d) 
Answer: (c)
[ The distance 
unit ]
Question: The distance of the point 
from X-axis is : [SEBA 2017]
(a) 2 (b) 5 (c) 1 (d) 3
Answer: (d) 3 units
[For the point P(2,3), the y-coordinate is 3. So, the distance from the X-axis is 3 units. ]
Question: The mid-point of the line segment joining the points 
and 
is 
, then 
is : [SEBA 2016]
(a) – 4 (b) – 12 (c) 12 (d) – 6
Answer: (b) – 12
[ We know that, the coordinates of the mid-point 
of the join of the points 
and 
is 
.
A/Q, 
]
Question: The distance between the point 
and 
is :
(a) 
(b) 
(c) 
(d) 
Answer: (c) 
.
[ Using the distance formula , we have
unit ]
Question: The distance between the points P
and Q
is : [ CBSE 2020 Basic]
(a) 
units (b) 
units (c) 
units (d) 40 units
Answer: (c) units
[ Using the distance formula , we have
units ]
Question: The point on the x-axis which is equidistant from 
and 
is :[ CBSE 2020 Standard ]
(a) 
(b) 
(c) 
(d) 
Answer: (d) .
[ Since the point 
is equidistant from and
So, 
Therefore, the point is 
. ]
Question: The centre of a circle whose end points of a diameter are 
and 
is : [ CBSE 2020 Standard ]
(a) 
(b) 
(c) 
(d) 
Answer: (c)
[ Let the point 
is the midpoint of and .
A/Q , 
]
Question: If the points 
and 
lie on the y-axis , then the distance of CD is :
(a) 
units (b) 8 units (c) 3 units (d) 5 units
Answer: (d) 5 units .
[Since, Both points C(0,3) and D(0,8) lie on the Y-axis.
Using distance formula, 
units ]
Question: The coordinates of the point which divides the join of 
and 
in the ratio 
is :
(a) (3 , 7) (b) (7 , 3) (c) (4, 8) (d) 
Answer: (b) (7, 3)
[ Here, 
, 
,
, 
, 
and 
and 
]
Question: If the coordinates of one end of a diameter of a circle are 
and the coordinates of its centre are 
, then the coordinate of the other end of the diameter is : [CBSE 2012]
(a) (1 , 6) (b) (6 , 1) (c) (1, 5) (d) 
Answer: (b) (6 , 1)
[ Since the point is the midpoint of and .
A/Q , 
and 
and 
]
Question: The point which lies on the perpendicular bisector of the line segment joining the points 
and 
is :
(a) (1 , 2) (b) (0 , 3) (c) (2, 0) (d) (0 , 0)
Answer: (d) (0 , 0)
[ Let, the point 
is the midpoint of 
and 
.
A/Q , 
Question: If P and Q be the points of trisection of the line segment joining the points 
and 
such that P is nearer to A , then the ratio of 
(a) 1 : 2 (b) 1 : 3 (c) 2 : 1 (d) 2 : 3
Answer: (c) 2 : 1
[ Since, P and Q trisect AB, so Q divides AB in ratio AQ : QB = 2 : 1 . Hence AQ : BQ = 2 : 1 .]
Question: If the distance between the points 
and 
is 5,then 
is :
(a) 5 (b) 0 (c) – 5 (d) 1
Answer: (b) 0
[ Since the distance between the points 
and 
is 5 .
A/Q , 
]
Question: The line segment joining 
and 
is divided by the -axis , then the ratio is :
(a) 1: 2 (b) 2 : 1 (c) 1 : 1 (d) 1 : 3
Answer: (c) 1 : 1
[ Given, -axis , i.e., 
.
We have , 
]
Question: If the points 
are collinear , then the value of is : [ CBSE 2014]
(a) 62 (b) – 63 (c) – 60 (d) 68
Answer: (b) – 63
[ Here , 
, 
, 
, 
, 
, 
We know that , 
]
Answers of the following Questions
Question: Find the mid-point of the line segment joining the points 
and 
.
Solution: We know that, 
Therefore, the mid-point of the line segment is 
.
Q2. If 
is the mid-point of the line segment joining 
and 
, then find 
.
Solution: Here , 
, 
, 
, 
, 
, 
We know that , 
Question: Find the distance of a point 
from the origin .
Solution: Given , the distance of the point 
and 
is
Question: Find the distance between the points 
and 
.
Solution: Given , the distance of the point 
and 
is
units
Question: If the distance between the points 
and is 5 , then find the value of 
. [CBSE 2017 ]
Solution: Given, the distance between the points 
and 
is 5 .
A/Q , 
SECTION = B
Question: Check whether 
and 
are the vertices of an isosceles triangle .
Solution : Let 
and 
are the vertices of any triangle respectively .
Using distance formula , we have
units
units
units
So, 
Therefore , the points 
and 
are the vertices of an isosceles triangle .
Question: Find the perimeter of a triangle with vertices 
, and . [CBSE 2014 F]
Solution: Let 
,
and 
are the vertices of the triangle.
The perimeter of 
units
Question: Find the point on the -axis which is equidistant from 
and 
. [SEBA19]
Solution: Let 
is equidistant from 
and R .
A/Q, 
Therefore , the point is 
.
Question: The -coordinate of a point P is twice its 
-coordinate . If P is equidistant from and 
, find the coordinates of P . [2016D]
Solution: Let the coordinate of the point P is .
Given, the -coordinate of a point P is twice its -coordinate , i.e., 
.
A/Q, 
.
So, the coordinate of the point P is 
.
Question: Find the values of 
for which the distance between the points 
and
is 10 units .
Solution : We have , 
[ using distance formula]
or 
Thus, the value of are – 9 and 3 .
Question: Prove that the points 
, 
and are the vertices of a right angled isosceles triangle . [CBSE 2016]
Solution: Let, the points 
, 
and 
are the vertices of the triangle.
Using Distance formula,
units
units
units
. So, ABC is an isosceles triangle .
Therefore, 
units and 
units
So, ABC is a right angled isosceles triangle . Proved.
Question: If 
, 
, 
and 
are the vertices of a parallelogram taken in order, find and .
Solution: Let, 
, 
, 
and 
are the vertices of a parallelogram respectively.
We know that diagonals of a parallelogram bisect each other .
Therefore, the coordinates of the mid-point of AC = the coordinates of the mid-point of BD .
and 
and 
and 
Question: Find the value of 
if the points 
, 
and 
are collinear .
Solution: Here , 
, 
, 
, 
, 
, 
We know that ,
Therefore, the value of k is 0 .
Question: If the distance of from 
and 
are equal , then prove that 
.
Solution: Given , the distance of from and are equal .
A/Q , 
Proved.
Question: Find the ratio in which the line segment joining 
and
is divided by the 
-axis . Also , find the coordinates of the point of division .
Solution: let , the ratio be 
and the coordinate is 
.
Here, 
, 
Using section formula , we have
and
Now,
Again, 
Therefore, the ratio is 1 : 1 and the coordinate is 
.
Question: If the point is equidistant from the points 
and 
,prove that 
. [CBSE 2017C]
Solution: Since, 
is equidistant from the points A and B 
.
A/Q , 
Proved.
Question: Find a relation between and such that the point is equidistant from the point 
and 
.
Solution: Given ,the point 
is equidistant from the point 
and 
.
A/Q , 
[ Squaring both side]
Question: Find the ratio in which the line segment joining the points 
and 
is divided by 
.
Solution: Let , the ratio be .
Here, 
, 
, 
Using section formula , we have
and
Now,
Therefore, the ratio is 2 : 7 .
SECTION = C
Question: If 
is equidistant from 
and 
, find the values of 
. Also find the distance QR and PR .
Solution : Since is equidistant from and 
.
A/Q, 
The distance of and 
is
units
The distance of and 
is
units
The distance of and is
units
The distance of and is
units
Question: If A
, 
, C 
and D
are the vertices of a quadrilateral ,find the area of the quadrilateral ABCD .
Solution: Since A , 
, 
and 
are the vertices of quadrilateral ABCD .
Join BD and we find ABD and BCD are two triangle .
sq. unit
and 
sq. unit
Therefore, area of 
Sq.units 
sq. unit
Question: If 
, 
and 
are the vertices of a right angled triangle with
, then find the value of 
. [Delhi 2015]
Solution: Since, , and are the vertices of a right angled triangle .
∴ 
Therefore, the value of t is 1 .
Question: Find the ratio in which the point 
divides the line segment joining the points 
and 
. Also , find the value of 
. [CBSE 2016]
Solution: Let, the ratio is 
.
Here, 
, 
, 
, 
, 
and 
Using section formula , we have
and 
Therefore, the ratio is 2 : 1 and the value of is 
.
Question: Find the coordinates of a point A , where AB is the diameter of a circle whose centre is 
and 
is 
.
Solution: Here, AP = BP = Radius . So, 
Let the coordinates of a point 
.
Here, 
, 
Using the section, we have
and
Therefore, the coordinate of the point 
.
Question: Find the ratio in which the 
-axis divides the line segment joining the points 
and 
. Also, find the point of intersection .
Solution: Let the ratio is and the coordinates of the point is 
.
Here , 
, 
, 
, 
Using section formula , we have
and 
Therefore, the ratio is 5 : 1 and the coordinates of the point is 
.
Question: If A 
, B 
, C
and D are the vertices of a parallelogram ABCD, find the values of 
and 
. Hence find the lengths of its sides . [CBSE 2018]
Solution: We know that diagonals of a parallelogram bisect each other .
So, the coordinates of the mid-point of AC = the coordinates of the mid-point of BD .
Thus, A , B , C
and D are the vertices of a parallelogram ABCD .
units
units
units
units
Question: Find the ratio in which the point of intersection of the -axis and the line segment which joins the points and 
internally divides the line segment . Also, find the coordinates of the point . [SEBA 2014]
Solution: let the ratio is and the coordinates of the point is .
Here , 
,
, 
, 
Using section formula , we have
and 
Therefore, the ratio is 5 : 7 and the coordinates of the point is 
.
Question: If A and B are 
and 
respectively , find the coordinates of P such that 
and P lies on the line segment AB .
Solution: Let, the coordinate of P is 
Given,
Here, , 
, 
, 
, 
and
Using section formula, We have
and 
Therefore, the coordinate of P is 
.
Question: Determine the ratio in which the line 
divides the line segment joining the points A
and B 
. Also, find the coordinate of the points .
Solution: let the ratio is and given the coordinate is .
Here, , , 
and 
Using section formula, We have
and 
A/Q , 
and 
Therefore, the ratio is 2 : 9 and the coordinate of the point is 
.
Question: If A , B
, C
and D
are the vertices of a quadrilateral ABCD of area 80 Square units , then find positive value of .
Solution: Given, ABCD be a quadrilateral and BD join .
ar(
) 
Sq. units (positive value)
and ar(
) 
sq. units (positive value)
A/Q , 
Therefore, the positive value of is 8 .
Question: Show that the points 
, 
, 
and 
are the vertices of a square .
Solution: Let, 
and 
are the given points.
Using distance formula,
and 
.
Therefore, 
is a square .
SECTION = D
Question: Find the coordinates of the points which divide the line segment joining 
and 
into four equal parts .
Solution: Let the coordinate of the points are 
and 
.
Here, 
, 
For point P : 
Using section formula , We have
and 
The coordinate of the point P is 
.
For point Q : Here, 
and 
The coordinate of the point Q is (0 , 5) .
For point R : Here, 
and 
The coordinate of the point R is 
.
Question: In figure 6, ABC is a triangle coordinates of whose vertex A is 
. D and E respectively are the mid-points of the sides AB and AC and their coordinates are and 
respectively . If F is the mid-point of BC , find the areas of 
and 
.
Solution: let 
and 
are the coordinates of the triangle ABC .
Since D be a mid-point of AB . So , 
and 
Since E be a mid-point of AC . So , 
and 
Therefore, the coordinates of triangle B and C are 
and 
.
The vertices of the triangle ABC are 
, 
and 
.
sq. unit
Again , F is a mid-point of BC , then the coordinate of F is 
,i.e. (1 , 2) .
The vertices of the triangle DEF are 
,
and 
.
sq. unit ( Area always positive)
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