• Dispur,Guwahati,Assam 781005
  • mylearnedu@gmail.com

7. COORDINATES GEOMETRY (SCERT)

SEBA Class 10 Maths Chapter 7. COORDINATES GEOMETRY

Chapter 7. Coordinate Geometry

Chapter 7 . Coordinate Geometry

Exercise 7.1 complete solution

Exercise 7.2 complete solution

Exercise 7.3 complete solution

Exercise 7.4 (Optional*) complete solution

 Distance formula :

1. The distance between  and  is given by  
2. The distance of a point from the origin  is given by  

  Section formula :
3. The coordinates of the point  which divides the line segment joining the points  and  internally in the ratio  are given by 

i.e. ,    and  
4. The mid-point of the line segment joining the points  and  is given by

i.e.,   and 

5. If ,  and  are the vertices of  , then the coordinates of the centroid is

 

 Area of a Triangle :

6. The area of the triangle formed by the points ,  and  is given by 

7.  The points ,  and  are collinear ,then

       

Class 10 Maths Chapter 7. Coordinate Geometry Exercise 7.1 Solutions

1. Find the distance between the following pairs of points :

   (i)       

   (ii)    

   (iii)   

Solution :  (i)       

Let  and  are two points .

Using distance formula , we have

 

     units   

(ii)      

Let and  are two points .

Using distance formula , we have

 

  

   units   

(iii)

Let  and are two points .

Using distance formula , we have

  

   units   

2. Find the distance between the points and   . Can you now find the distance between the two towns A and B discussed in figure 7.2 .

Solution : Let   and  are two points .

  Using distance formula , we have

    

    

    

     units 

Second part :

Here, the point A and B lie on the x-axis .

Iin figure , OA = 4 units and OB = 6 units

AB = OB - OA = 6 - 4 = 2 units

3. Determine if the points   and  are collinear .

Solution : Let   and are three points respectively .

Using distance formula , we have

  

    units   

    units   

 

  units   

So ,  .

Therefore, the points   and  are not collinear .

4. Check whether 5 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 .

5. In a classroom, 4 friends are seated at the points  A , B , C and D as shown in Fig. 7.8 . Champa and Chemeli 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 .

Solution: Given, the coordinates of the four friends are A(3,4) , B(6,7) , C(9,4) and D(6,1) respectively .

In figure :

           

Using distance formula, we have

  units   

 units   

 units   

Again, 

 units   

And

 units   

So, AB = BC = CD = AD and AC = BD

Therefore, ABCD is a square . Chapma is correct .

6. Name the type of quadrilateral formed, if any , by the following points, and give reasons for your answers :

(i)     

(ii)

(iii)  

Solution:  (i)    

 Let and  are the vertices of the quadrilateral respectively.

Using distance formula , we have

 units

 units

 units

 units

 units

 units

     

So,  and  

Therefore , and  are vertices of the square . 

(ii)

Solution:  Let  and are the vertices of the quadrilateral respectively.

Using distance formula , we have

 units

 units

 units

 units

So,

Therefore ,  and  are not of the vertices of  quadrilateral . 

(iii)  

Solution: Let  and   are the vertices of the quadrilateral respectively.

Using distance formula , we have

 units

 units

 units

 units

 units

 units

So, ,  and

    

Therefore , and  are vertices of the parallelogram . 

7. Find the point on the -axis which is equidistant from  and  .

Solution : Let,  is equidistant from the points A(2 , – 5 ) and B (– 2 ,9) .

Given , -axis , i.e., .  

  A/Q ,  

      

      

     

    

    

    

   

    Therefore , the coordinate of the point P is  .

8. Find the values of  for which the distance between the points  and is 10 units .

Solution :  In given figure :

We have ,  

 

 

 

 

 

 

 

 

 

  or 

  Thus, the value of  are – 9  and 3 .

9. 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

10. 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  .

We have,     

  

    [Squaring both side]

 

     

  

  

 

 

   

Class 10 Maths Chapter 7. Coordinate Geometry Exercise 7.2 Solutions

1. Find  the coordinates of the point which divides the join of   and  in the ratio  .

Solution:  Here, , ,  

Let the coordinate of the point is P .

Using section formula , we have

And 

Therefore, the coordinates of the point is (1 , 3) .

2. Find the coordinates of the points of trisection of the line segment joining  and  .

Solution:  let the coordinates of the points are  and Q .

For  point P : Here,  , ,  

Using section formula , we have

And  

For point Q : Here, , ,

And 

Therefore, the coordinates of the points are and  .

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 flowers pots have been placed at a distance of 1 m from each other along AD , as shown in Fig. 7.12 . Niharika runs  th the distance AD , on the  line and posts a green flag . Preet runs  th 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 jointing the two flags, where should she post her flag ?

Solution:  Given, ABCD is a rectangular school ground , then the distance of the side AD = 1m × 100 = 100 m .

The distance of AD run by Niharika on the second line

Therefore, the coordinate of the point is (2 , 25) .

 Again , the distance of AD run by Preet on the eighth line

Therefore, the coordinate of the point is (8 , 20) .

Using distance formula, we have

The distance between both the flags

m

Since, Rashmi has to post a blue flag exactly halfway between the line segment jointing the two flags ,i.e., Rashmi is equidistant from Niharika and Preet . let the coordinate of Rashmi is

Using section formula , we have

 and 

Therefore, the position of Rashmi has to post a blue flag on the 5th line at a distance of 22.5 m .

4. 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 .

5. 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  .

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

Solution:  We know that , the diagonals of a parallelogram bisect each other .

A/Q, The coordinate of the mid-point of the diagonal AC = The coordinate of the mid-point of the diagonal BD .

Or  

Therefore, the value of  and  .

7. 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 A is  .

Here, , 

Using the section, we have

 

and  

Therefore, the coordinate of the point  .

8. 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 coordinates of the point P is  .

Now,  

Here, ,,  

Using section formula , we have

 

and 

Therefore, the coordinate of the point P is  .

9. 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 .

10. Find the area of a rhombus if its vertices are  and  taken in order . [Hint : Area of a rhombus (product of its diagonals)]

Solution:  let the vertices of the rhombus are  and  respectively .

Using distance formula , we have,

 units

and

 units

The area of a rhombus ABCD

sq. units

Class 10 Maths Chapter 7. Coordinate Geometry Exercise 7.3 Solutions

1. Find the area of the triangle whose vertices are :

   (i)       (ii)  

Solution: (i)         

 let A(2,3) , B(– 1 , 0) and C(2 ,– 4) are the vertices of the triangle ABC respectively .

Here, ,

We know that ,

Area of

    Sq. units .

Solution: (ii) 

 let A(– 5 ,– 1) , B(3 ,– 5) and C(5 , 2) are the vertices of the triangle ABC respectively .

Here, ,

We know that ,

Area of

Sq. units

2. In each of the following find the value of ‘k ’ , for which the points are collinear  :

   (i)         (ii) 

Solution:   (i)        

Here, ,

We know that,

Therefore, the value of k is 4 .

Solution: (ii)  

Here, ,

We know that,

Therefore, the value of k is 3 .

3. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are and .Find the ratio of this area to the area of the given triangle .

Solution:  let A(0 ,– 1) , B(2 , 1) and C(0 , 3) are the vertices of the triangle ABC respectively .

Here, ,

We know that ,

Area of

sq. units

     

Let D , E and F are the mid-point of the sides AB , BC and AC of the triangle ABC respectively .

So, the coordinate of the point D is

The coordinate of the point E is

The coordinate of the point F is

Therefore, D(1 , 0) , E(1 , 2) and C(0 , 1) are the vertices of the triangle DEF respectively .

Here, ,

Area of

sq. units

So,

4. Find the area of the quadrilateral whose vertices, taken in order are  and .

Solution: let A(– 4 ,– 2) , B(– 3 , – 5) , C(3 ,– 2) and D(2,3) are the vertices of the triangle ABC respectively and join diagonal AC .Then ,we find two triangles ABC and ADC .

  

For triangle ABC : The vertices of the triangle ABC are A(– 4 ,– 2) , B(– 3 , – 5) and C(3 ,– 2) respectively.

 Here, ,

We know that ,

Area of

Sq. units

For triangle ADC : The vertices of the triangle ABC are A(– 4 ,– 2) , D(2 , 3) and C(3 ,– 2) respectively.

 Here, ,

We know that ,

Area of

Sq. units (positive value)

Therefore, Area of ABCD = Area of ABC + Area of ADC

Sq.units .

5. You have studied in Class IX , (Chapter 9 , Example 3) , that a median of a triangle divides it into two triangles of equal areas . Verify this result for  whose vertices are  and  .

Solution:  let AD is the median of a triangle ABC . The vertices of the triangle ABC are  and  . We find two triangles are ABD and ACD .

  

The coordinate of the point D is

For triangle ABD : The vertices of the triangle ABD are A( 4 ,– 6) , B(3 ,– 2) and D(4 ,0) respectively.

 Here, ,

We know that ,

Area of

Sq. units (positive value only)

For triangle ACD : The vertices of the triangle ABD are A( 4 ,– 6) , B(5 , 2) and D(4 ,0) respectively.

 Here, ,

We know that ,

Area of

Sq. units

Therefore,  area of the ABD = Area of the ACD . Verified

Class 10 Maths Chapter 7. Coordinate Geometry Exercise 7.4 (Optional)* Solutions

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).

Solution: let , the ratio is  

 Here, ,

Using section formula , we have

and  

Since the line  passed through the point P() , then

Therefore, the ratio is 2 : 9 .

2. Find a relation between  and  if the points (), (1, 2) and (7, 0) are collinear.

Solution:  Here,

We know that,

 

3. Find the centre of a circle passing through the points (6, – 6), (3, – 7) and (3, 3).
4. The two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of the other two vertices.
5. The Class X students of a secondary school in Krishinagar have been allotted a rectangular plot of land for their gardening activity. Sapling of Gulmohar are planted on the boundary at a distance of 1m from each other. There is a triangular grassy lawn in the plot as shown in the Fig. 7.14. 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?
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 Theorem 6.2 and Theorem 6.6).
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 the point D.
(ii) Find the coordinates of the point P on AD such that AP : PD = 2 : 1
(iii) Find the coordinates of points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1 .
(iv) What do yo observe?
[Note : The point which is common to all the three medians is called the centroid and this point divides each median in the ratio 2 : 1.]

(v) If A( ) , B( ) and C( ) are the vertices of ∆ ABC, find the coordinates of the centroid of the triangle.
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 a square? a rectangle? or a rhombus? Justify your answer.