7. COORDINATES GEOMETRY (NCERT)

CBSE Class 10 Chapter 7. COORDINATES GEOMETRY (NCERT)

Chapter 7. Coordinate Geometry

Chapter 7 . Coordinate Geometry

Exercise 7.1 complete solution

Exercise 7.2 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

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

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

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

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

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

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

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 ,

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

units

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 .

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