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

(i) (2, 3), (4, 1)

(ii) (-5, 7), (-1, 3)

(iii) (a, b), (- a, – b)

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Distance formula =

(i) Applying Distance Formula to find distance between points (2, 3) and (4,1), we get

d =

(ii) Applying Distance Formula to find distance between points (–5, 7) and (–1, 3), we get

d =

(iii) Applying Distance Formula to find distance between points (a, b) and (–a, –b), we get

d =

2. Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

The coordinates of point A are (0, 0) and coordinates of point B are (36, 15).

To find the distance between them, we use Distance formula:

d =

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let A = (1, 5), B = (2, 3) and C = (–2, –11)

Using Distance Formula to find distance AB, BC and CA.

AB =

BC =

CA =

Since AB + AC

Therefore, the points A, B and C are not collinear.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let A = (5, –2), B = (6, 4) and C = (7, –2)

Using Distance Formula to find distances AB, BC and CA.

AB =

BC =

CA =

Since AB = BC.

Therefore, A, B and C are vertices of an isosceles triangle.

5. In a classroom, 4 friends are seated at the points A (3, 4), B (6, 7), C (9, 4) and D (6, 1). Champa and Chameli 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.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

We have A = (3, 4), B = (6, 7), C = (9, 4) and D = (6, 1)

Using Distance Formula to find distances AB, BC, CD and DA, we get

AB =

BC =

CD =

DA =

Therefore, All the sides of ABCD are equal here. … (1)

Now, we will check the length of its diagonals.

AC =

BD =

So, Diagonals of ABCD are also equal. … (2)

From (1) and (2), we can definitely say that ABCD is a square.

Therefore, Champa is correct.

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

(i) (- 1, – 2), (1, 0), (- 1, 2), (- 3, 0)

(ii) (- 3, 5), (3, 1), (0, 3), (- 1, – 4)

(iii) (4, 5), (7, 6), (4, 3), (1, 2)

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

i) Let A = (–1, –2), B = (1, 0), C= (–1, 2) and D = (–3, 0)

Using Distance Formula to find distances AB, BC, CD and DA, we get

AB =

BC =

CD =

DA =

Therefore, all four sides of quadrilateral are equal. … (1)

Now, we will check the length of diagonals.

AC =

BD =

Therefore, diagonals of quadrilateral ABCD are also equal. … (2)

From (1) and (2), we can say that ABCD is a square.

(ii) Let A = (–3, 5), B= (3, 1), C= (0, 3) and D= (–1, –4)

Using Distance Formula to find distances AB, BC, CD and DA, we get

AB =

BC =

CD =

DA =

We cannot find any relation between the lengths of different sides.

Therefore, we cannot give any name to the figure ABCD.

(iii) Let A = (4, 5), B= (7, 6), C= (4, 3) and D= (1, 2)

Using Distance Formula to find distances AB, BC, CD and DA, we get

AB =

BC =

CD =

DA =

Here opposite sides of quadrilateral ABCD are equal. … (1)

We can now find out the lengths of diagonals.

AC =

BD =

Here diagonals of ABCD are not equal. … (2)

From (1) and (2), we can say that ABCD is not a rectangle therefore it is a parallelogram.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let the point be (x, 0) on x–axis which is equidistant from (2, –5) and (–2, 9).

Using Distance Formula and according to given conditions we have:

=>

Squaring both sides, we get

= >

=>

=>

=>

Therefore, point on the x–axis which is equidistant from (2, –5) and (–2, 9) is (–7, 0)

8. Find the values of y for which the distance between the points P (2, –3) and Q (10, y) is 10 units.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Using Distance formula, we have

=>

=>

Squaring both sides, we get

100 =

=>

Solving this Quadratic equation by factorization, we can write

=>

=>

=>

=>

9. If, Q (0, 1) is equidistant from P (5, –3) and R (x, 6), find the values of x. Also, find the distances QR and PR.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

It is given that Q is equidistant from P and R. Using Distance Formula, we get

PQ = RQ

PQ=

QR=

So,

On squaring both sides,

Coordinates of point R will be (4,6) or (-4,6),

If R is (4,6),

QR =

PR =

If R is (-4,6),

QR =

PR =

10. Find a relation between x and y such that the point (x, y) is equidistant from the point (3, 6) and (–3, 4).

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

It is given that (x, y) is equidistant from (3, 6) and (–3, 4).

Using Distance formula, we can write

=>

Squaring both sides, we get

=>

=>

=>

=>

1. Find the coordinates of the point which divides the join of (- 1, 7) and (4, – 3) in the ratio 2:3.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let the points be P(x,y), Using the section formula

Therefore the point is(1,3)

Q.2 Find the coordinates of the points of trisection of the line segment joining (4, -1) and (-2, -3).

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let P (x_{1}, y_{1}) and Q (x_{2}, y_{2}) are the points of trisection of the line segment joining the given points i.e., AP = PQ = QB

Therefore, point P divides AB internally in the ratio 1:2.

x_{1} =

y_{1} =

_{1}, y_{1}) = P(2,

Point Q divides AB internally in the ratio 2:1.

x_{2} =

y_{2} =

The coordinates of the point Q(x_{2} ,y_{2} ) = (0,

Q.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 flower pots have been placed at a distance of 1 m from each other along AD, as shown in the following figure. Niharika runs

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

The green flag is

Similarly coordinate of red flag = (8 , 20)

Distance between red and green flag

=

Now blue flag is posted at the mid point of two flag

Then let, the coordinate of blue flag = (x , y )

Q.4 Find the ratio in which the line segment joining the points (-3, 10) and (6, – 8) is divided by (-1, 6).

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let the required ratio be k : 1

By section formula ,

x =

==> -1 =

==> -k-1 = 6k -3

==> 7k = 2

==> k =

The required ratio = k : 1

Q.5 Find the ratio in which the line segment joining A (1, – 5) and B (- 4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let p(x , 0) be the point which devides the line segment joining A(1, -5 ) and B(-4,5 ) in the ratio m : 1

Then the using section formula

(x , 0 ) =

==> 0 =

==> 0 = 5m - 5

==> m = 5 /5 = 1

Hence the required ratio is 1 : 1

Since the ratio is 1 : 1 , so P is the mid point

x =

Q.6 If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Mid point of AC = Mid point BD

==>

==>

==> x +1 = 7

==> x = 7-1= 6

==>

==> y+5 = 8

==> y = 8 -5 = 3

Q.7 Find the coordinates of a point A, where AB is the diameter of circle whose centre is (2, – 3) and B is (1,4).

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let the coordinates of point A be (x, y).

Mid-point of AB is (2, – 3), which is the centre of the circle.

Coordinate of B = (1, 4)

(2, -3) =

and

The coordinates of A(3,-10).

Q.10 Find the area of a rhombus if its vertices are (3, 0), (4, 5), (-1, 4) and (-2,-1) taken in order.

[Hint: Area of a rhombus = 1/2(product of its diagonals)

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let A(3, 0), B (4, 5), C( – 1, 4) and D ( – 2, – 1) are the vertices of a rhombus ABCD.

Length of diagonal AC

Length of diagonal BD

Area of rhombus =

=

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

(i) (2, 3), (-1, 0), (2, -4)

(ii) (-5, -1), (3, -5), (5, 2)

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

We know that formula for area of a triangle whose vertices are

=

(i) So, here

So, area of triangle =

? Area of triangle is

(ii) Similarly, here

So, area of triangle =

? Area of triangle is

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

(i) (7, -2), (5, 1), (3, -k)

(ii) (8, 1), (k, -4), (2, -5)

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

We know that for collinear points area of triangle = 0 ,i.e.,

(i)

=>

=>

=>

(ii)

=>

=>

=>

3. Find the area of the triangle formed by joining the mid-points of the sides of the triangle whose vertices are (0, -1), (2, 1) and (0, 3). Find the ratio of this area to the area of the given triangle.

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

The vertices of the triangle ABC are A (0, -1), B (2, 1), C (0, 3).

Let D, E, F be the mid-points of the sides AB,BC,CA of this triangle ABC respectively.

Coordinates of D, E, and F are given by :

D =

E = (

F = (

So, Area of

? Area of ? DEF = 1 sq. units

Area of ? ABC =

? Area of ? ABC = 4 sq. units

So,

4. Find the area of the quadrilateral whose vertices, taken in order, are (-4, -2), (-3, -5), (3, -2) and (2, 3).

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Draw a line from B to D

Now we have,

Area of quad. ABCD = | Area of ? DAB | + | Area of ? BCD |

So,

Area of ? DAB =

Similarly, Area of ? BCD =

? Area of quad. ABCD =

5. You have studied in Class IX that a median of a triangle divides it into two triangles of equal areas. Verify this result for ?ABC whose vertices are A (4, – 6), B (3, – 2) and C (5, 2).

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

The vertices of triangle ABC are A(4,-6), B (3, – 2) and C (5, 2).

Let D be the mid-point of side BC of ? ABC. Therefore, AD is the median in ? ABC.

So, coordinates of D are (

Therefore, coordinates of D = (4,0)

Now,

Area of ? ABD =

=

=

Similarly,

Area of ? ACD =

=

=

? Clearly, Area of ? ABD = Area of ? ACD = 3 sq. units.

Hence, median of a triangle divides it into two triangles of equal areas.

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).

NCERT Solutions for Class 10 Maths Chapter 7 Coordinate Geometry

Answer :

Let the line 2x + y - 4 = 0 divides AB in 1 : k ratio, then the coordinates of the point of division is

The point of intersection lie on both lines

?

=>