Chapter 7.  COORDINATES  GEOMETRY

Class 10 Maths Chapter 6 Coordinates Geometry : Multiple Choice Questions , Answer following the Questions , Fill in the blanks , 2 Marks Questions , 3 Marks Question , 4 Marks and Solutions :    

 Class 10 Coordinate Geometry Multiple Questions and Answers

SECTION = A

Q1. The point  is equidistant from the points  and  ,then : [ SEBA 2020]

(a)                                                               

(b) 

(c)                                                                

(d)  

Solution:  (c)   .

[ let the distance of   from   and   are equal .

A/Q,   

 

 

 

 

 

       

       

               ]

Q2. The distance between the points  and  is :     [SEBA 2019]

(a)  10 units               

(b)  8 units                  

(c)  6  units                 

(d)  2 units

Solution:    (b)  10                               

[   The distance between the points

   units  ]

Q3.  If   is mid-point of the line segment joining the points  and  ,then  is :

 (a)   – 1       (b)  1      (c)  2          (d)  – 2

Solution:  (b)  1                 

[ We have ,    

  

 

 

     ]

Q4. The distance between the points  and   is   :      [SEBA 2015 , 18]

 (a)  2       (b)          (c)   1       (d)  0

Solution:    (b)                                  

 [ The distance between the points

    unit    ]

Q5. The line segment joining the points  and  in the ratio 2 : 3 , then the coordinate of the point is : 

  (a)          (b)         (c)         (d) 

Solution:   (d)   .

[  Here,  ,  ,    ,   ,    and  

 

    

 and 

 

   ]

Q6. The distance between the point  and  is :

 (a)          (b)           (c)         (d) 

Solution:  (c)     

 [ The distance  

   unit ]

Q7. The distance of the point  from X-axis is :       [SEBA 2017]

(a)  2     (b)  5        (c)  1       (d)  3

Solution:   (d) 3 units

Q8. The mid-point of the line segment joining the points and  is  , then  is :   [SEBA 2016]

(a)   – 4          (b)   – 12       (c)  12        (d)   – 6   

Solution:  (b) – 12                   

   [ We know that, the coordinates of the mid-point  of the

    join of the points and  is  .

 A/Q,    

  

  

 

  

Q9. The distance between the point  and  is :

(a)            

(b)         

(c)        

(d) 

Solution:  (c)    .   

[ Using the distance formula , we have

 

 

 

  unit ]

 Q10. The distance between the points P and Q is : [ CBSE 2020 Basic]

 (a)   units              (b)  units              (c)     units              (d) 40 units

Solution:  (c)    units           

[    Using the distance formula , we have

 

 units     ]

Q12. The point on the x-axis which is equidistant from  and  is :[ CBSE 2020 Standard ]

(a)         (b)         (c)          (d)   

Solution:    (d)    .

 [ Since the point  is equidistant from  and  

 So,       

 Therefore, the point is  . ]

Q13. The centre of a circle whose end points of a diameter are   and  is : [ CBSE 2020 Standard ]

    (a)              (b)           (c)            (d)  

Solution:   (c)      .                                  

[ let the point  is the midpoint of   and  .

A/Q ,           ]

Q14. 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

Solution:   (d)  5 units .

[  The distance  units ]   

 Class 10 Chapter 7. Coordinate Geometry  Filled in the blanks

Q1. The coordinates of the point which divides the join of and  in the ratio  is  .

Solution:       .

  [ Here,   ,   , ,  ,  and   

       

 and          ]

Q2. 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]

Solution:    .                                             

[ Since the point  is the midpoint of   and  .

 A/Q ,     

             and    

              

                                  ]

Q3. The point which lies on the perpendicular bisector of the line segment joining the points  and  is  .

Solution:    .              

[ let  the point  is the midpoint of   and  .

A/Q ,     

Q4. 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   .

Solution:   2 : 1

[ The ratio of the points  and  is  2 : 1 .  ]

Q5. If the distance between the points  and  is 5,then  is .

Solution:   0       

[ Since the distance between the points  and  is 5 .

 A/Q ,    

 

 

 

 

      ]

 Q6. The line segment joining  and  is divided by the -axis , then the ratio is   .

Solution:     1 : 1      

 [ Given, -axis , i.e.,  . 

 A/Q,       

 

 

 

 

   ]

Q7.  If the points   are collinear , then the value of  is  .   [ CBSE 2014]

Solution:    – 63   

 [  Here ,  ,    ,  ,    ,    ,  

We know that ,  

  

  

       ]

 Class 10 Coordinate Geometry Answers of the following Questions

Q1. 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 ,      

 

 

 

 

  

  

    

Q3. Find the distance of a point  from the origin .

Solution:  Given , the distance of the point  and  is

  

 

Q4. Find the distance between the points  and  .

Solution: Given , the distance of the point  and  is

 

 

 

  units

Q5. 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 ,           

      

 

 

   

 

   

Class 10 Coordinate Geometry 2 Marks Questions and Answers

    SECTION = B

Q1. 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

Q3. Find the point on the -axis which is equidistant from  and  . [SEBA19]

Solution:  Let   is equidistant from  and  .

 According to question,  

 

 

 

 

   

 Therefore , the point is  .

Q4. 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  .

Q5. 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.

       units

  units

 units

   

 So, ABC is an isosceles triangle .

  Therefore,  units and   units

So, ABC is a right angled isosceles triangle .  Proved. 

Q6. If   , ,  and  are the vertices of a parallelogram taken in order, find  and  .

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 .

 

                 and      

                         

                                  

      and      

Q7. Find the value of  if the points  ,  and  are collinear .

Solution:   Here ,  ,   ,   ,   ,   ,

 We know that ,

  

  

  

 

Q8. If the distance of  from  and   are equal , then prove that  .

Solution: Given , the distance of   from  and  are equal .

 A/Q,  

  

  

  

 

 

    Proved.

Q9. 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.

Q10. 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]

 

 

   

 Class 10 Coordinate Geometry 3 Marks Questions and Answers 

   SECTION = C

Q1. 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

Q2. If   ,   and  are the vertices of a right angled triangle with , then find the value of   .  [Delhi 2015]

Solution:  Since, A  ,    and   are the vertices of a right angled triangle .

  

 

  

 

 

    

Q3. 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   .

Q4. Find the ratio in which the -axis divides the line segment joining the points and   . Also, find the point of intersection .

Solution:  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    .

Q5. 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 .

 

  

            

          and    

             

              

                       

    Thus,  A , B , C and D are the vertices of a parallelogram ABCD .

units

  units

     units

   units  

Q6. 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   .

Q7. 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  

        

  

 

 

 

    

 Here,  ,   ,    ,     ,    and  

   Using section formula, We have

       

 and     

       

 Therefore, the coordinate of P is    .

Q8. 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  ,       

 

  

 

 

 

 

    

Here, m = 2 , n = 9

      

and       

 Therefore, the ratio is 2 : 9  and the coordinate of the point is  .

Q9. 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 .

Q10. Show that the points  ,  ,  and  are the vertices of a square .

Solution:  let,  and  are the given points.

   So,    

               

          

          

             

        

             and    .

 Therefore, is a square .

 Class 10 Coordinate Geometry 4 Marks Questions and Solutions                                                     

SECTION = D

Q1. 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)