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

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

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

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.