##### Class 10^{th} Mathematics CBSE Solution

**Exercise 7.1**- Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii)…
- Find the distance between the points (0, 0) and (36, 15). Can you now find the…
- Determine if the points (1, 5), (2, 3) and (- 2, - 11) are collinear…
- Check whether (5, - 2), (6, 4) and (7, - 2) are the vertices of an isosceles…
- In a classroom, 4 friends areseated at the points A, B, C and D as shown in Fig.…
- Name the type of quadrilateral formed, if any, by the following points, and give…
- Find the point on the x-axis which is equidistant from (2, 5) and (2, 9).…
- Find the values of y for which the distance between the points P(2, 3) and Q…
- If Q(0, 1) is equidistant from P(5, -3) and R(x, 6), find the values of x. Also…
- Find a relation between x and y such that the point (x, y) is equidistant from…

**Exercise 7.2**- Find the coordinates of the point which divides the join of (-1, 7) and (4, -3)…
- Find the coordinates of the points of trisection of the line segment joining…
- To conduct Sports Day activities, in your rectangular shaped school ground ABCD,…
- Find the ratio in which the line segment joining the points (- 3, 10) and (6, -…
- Find the ratio in which the line segment joining A(1, 5) and B(4, 5) is divided…
- If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken…
- Find the coordinates of a point A, where AB is the diameter of a circle whose…
- If A and B are (- 2, - 2) and (2, - 4), respectively, find the coordinates of P…
- Find the coordinates of the points which divide the line segment joining A(- 2,…
- Find the area of a rhombus if its vertices are (3, 0), (4, 5), (- 1, 4) and (-…

**Exercise 7.3**- Find the area of the triangle whose vertices are: (i) (2, 3), (-1, 0), (2, - 4)…
- In each of the following find the value of k, for which the points are collinear…
- Find the area of the triangle formed by joining the mid-points of the sides of…
- Find the area of the quadrilateral whose vertices, taken in order, are (- 4, -…
- You have studied in Class IX, (Chapter 9, Example 3), that a median of a…

**Exercise 7.4**- Determine the ratio in which the line 2x+y-4 = 0 divides the line segment…
- Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are…
- Find the centre of a circle passing through the points (6, - 6), (3, - 7) and…
- The two opposite vertices of a square are (-1, 2) and (3, 2). Find the…
- The Class X students of asecondary school inKrishinagarhave been allotteda…
- The vertices of a ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to…
- Let A (4, 2), B(6, 5) and C (1, 4) be the vertices of ABC (i) The median from A…
- ABCD is a rectangle formed by the points A(-1, -1), B(- 1, 4), C(5, 4) and D(5,…

**Exercise 7.1**

- Find the distance between the following pairs of points: (i) (2, 3), (4, 1) (ii)…
- Find the distance between the points (0, 0) and (36, 15). Can you now find the…
- Determine if the points (1, 5), (2, 3) and (- 2, - 11) are collinear…
- Check whether (5, - 2), (6, 4) and (7, - 2) are the vertices of an isosceles…
- In a classroom, 4 friends areseated at the points A, B, C and D as shown in Fig.…
- Name the type of quadrilateral formed, if any, by the following points, and give…
- Find the point on the x-axis which is equidistant from (2, 5) and (2, 9).…
- Find the values of y for which the distance between the points P(2, 3) and Q…
- If Q(0, 1) is equidistant from P(5, -3) and R(x, 6), find the values of x. Also…
- Find a relation between x and y such that the point (x, y) is equidistant from…

**Exercise 7.2**

- Find the coordinates of the point which divides the join of (-1, 7) and (4, -3)…
- Find the coordinates of the points of trisection of the line segment joining…
- To conduct Sports Day activities, in your rectangular shaped school ground ABCD,…
- Find the ratio in which the line segment joining the points (- 3, 10) and (6, -…
- Find the ratio in which the line segment joining A(1, 5) and B(4, 5) is divided…
- If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken…
- Find the coordinates of a point A, where AB is the diameter of a circle whose…
- If A and B are (- 2, - 2) and (2, - 4), respectively, find the coordinates of P…
- Find the coordinates of the points which divide the line segment joining A(- 2,…
- Find the area of a rhombus if its vertices are (3, 0), (4, 5), (- 1, 4) and (-…

**Exercise 7.3**

- Find the area of the triangle whose vertices are: (i) (2, 3), (-1, 0), (2, - 4)…
- In each of the following find the value of k, for which the points are collinear…
- Find the area of the triangle formed by joining the mid-points of the sides of…
- Find the area of the quadrilateral whose vertices, taken in order, are (- 4, -…
- You have studied in Class IX, (Chapter 9, Example 3), that a median of a…

**Exercise 7.4**

- Determine the ratio in which the line 2x+y-4 = 0 divides the line segment…
- Find a relation between x and y if the points (x, y), (1, 2) and (7, 0) are…
- Find the centre of a circle passing through the points (6, - 6), (3, - 7) and…
- The two opposite vertices of a square are (-1, 2) and (3, 2). Find the…
- The Class X students of asecondary school inKrishinagarhave been allotteda…
- The vertices of a ABC are A(4, 6), B(1, 5) and C(7, 2). A line is drawn to…
- Let A (4, 2), B(6, 5) and C (1, 4) be the vertices of ABC (i) The median from A…
- ABCD is a rectangle formed by the points A(-1, -1), B(- 1, 4), C(5, 4) and D(5,…

###### Exercise 7.1

**Question 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)

**Answer:**(i) We know that distance between the two points is given by:

d = √[(x_{1} – x_{2})^{2} + (y_{1} – y_{2})^{2}]

Distance between A(2, 3) and B(4, 1) is:

f =√ [(4 - 2)^{2} + (1 – 3)^{2}]

= √[4 + 4]= √8

= 2√2

(ii) Distance between A(-5, 7) and B(-1, 3) is:

f = √[(-5 + 1)^{2} + (7 – 3)^{2}]

= √[16 + 16] = √32

= 4√2

(iii) Distance between A(a, b) and B(-a, -b) is:

f = √[(a - (- a))^{2} + ( b - (-b))^{2}]

f = √[(a+ a)^{2} + (b+ b)^{2}]

f = √[2a + 2b] = √(4a^{2} + 4b^{2})

f = 2√(a^{2} + b^{2})

**Question 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

**Answer:**

**To find: Distance between two points**

**Given: Points A(0, 0), B(36, 15)**

For two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}),

Distance is given by

f = [(x_{2 - }x_{1})^{2} + (y_{2 - }y_{1})^{2}]^{1/2}

Distance between (0, 0) and (36, 15) is:

f = [(36 – 0)^{2} + (15 – 0)^{2}]^{1/2}

= [1296 + 225]^{1/2} = (1521)^{1/2}

= 39

Hence Distance between points A and B is 39 units

Yes, we can find the distance between the given towns A and B. Let us take town A at origin point (0, 0)

Hence, town B will be at point (36, 15) with respect to town A

And, as calculated above, **the distance between town A and B will be 39 km**

**Question 3.**Determine if the points (1, 5), (2, 3) and (– 2, – 11) are collinear.

**Answer:**Let the points (1, 5), (2, 3), and (- 2,-11) be representing the vertices A, B, and C of the given triangle respectively.

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

Case 1)

Since AB + BC ≠ CA

Case 2)

Now,

BA + AC ≠ BC

Case 3)

Now,

As BC+CA≠BA

As three of the cases are not satisfied.

Hence the points are not collinear.

**Question 4.**Check whether (5, – 2), (6, 4) and (7, – 2) are the vertices of an isosceles triangle

**Answer:**

Distance between two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given by

D = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Let us assume that points (5, - 2), (6, 4), and (7, - 2) are representing the vertices A, B, and C of the given triangle respectively as shown in the figure.

AB = [(5- 6)^{2} + (-2- 4)^{2}]^{1/2}

=

=

BC = [(6- 7)^{2} + (4 + 2)^{2}]^{1/2}

=

=

CA = [(5- 7)^{2} + (-2+ 2)^{2}]^{1/2}

=

= 2

Therefore, AB = BC

As two sides are equal in length, therefore, ABC is an isosceles triangle

**Question 5.**In a classroom, 4 friends areseated at the points A, B, C and D as shown in Fig. 7.8 Champaand Chameli walk into the classand after observing for a fewminutes Champa asks Chameli,“Don’t you think ABCD is asquare?” Chameli disagrees.Using distance formula, findwhich of them is correct

**Answer:**It can be seen that A (3, 4), B (6, 7), C (9, 4), and D (6, 1) are the positions of 4 friends

distance between two points A(x_{1, }y_{1}) and B(x_{2}, y_{2}) is given by

D = √(x_{2} - x_{1})^{2 }+ (y_{2 }- y_{1})^{2}

Hence,

AB = [(3- 6)^{2} + (4 - 7)^{2}]^{1/2}

=

=

BC = [(6 - 9)^{2} + (7- 4)^{2}]^{1/2}

=

=

CD = [(9- 6)^{2} + (4- 1)^{2}]^{1/2}

=

=

AD = [(3 - 6)^{2} + (4 - 1)^{2}]^{1/2}

=

=

Diagonal AC = [(3 - 9)^{2} + (4 - 4)^{2}]^{1/2}

=

= 6

Diagonal BD = [(6- 6)^{2} + (7- 1)^{2}]^{1/2}

=

= 6

It can be seen that all sides of quadrilateral ABCD are of the same length and diagonals are of the same length

Therefore, ABCD is a square and hence, Champa was correct

**Question 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)

**Answer:****To Find: Type of quadrilateral formed**

(i) Let the points ( - 1, - 2), (1, 0), ( - 1, 2), and ( - 3, 0) be representing the vertices A, B, C, and D of the given quadrilateral respectively

The distance formula is an algebraic expression used to determine the distance between two points with the coordinates (x_{1}, y_{1}) and (x_{2}, y_{2}).

D = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}AB = [(-1- 1)^{2} + (-2- 0)^{2}]^{1/2}

= =

= 2√2

BC = √[(1 + 1)^{2} + (0- 2)^{2}]

= =

= 2√2

CD = √[(-1 + 3)^{2} + (2- 0)^{2}]

= =

= 2√2

AD = √[(-1+ 3)^{2} + (-2- 0)^{2}]

= =

= 2√2

Diagonal AC = √[(-1 + 1)^{2} + (-2 - 2)^{2}]

=

= 4

Diagonal BD = √[(1 + 3)^{2} + (0- 0)^{2}]

=

= 4

It is clear that all sides of this quadrilateral are of the same length and the diagonals are of the same length. Therefore, the given points are the vertices of a square

(ii)Let the points (- 3, 5), (3, 1), (0, 3), and ( - 1, - 4) be representing the vertices A, B, C, and D of the given quadrilateral respectively

The distance formula is an algebraic expression used to determine the distance between two points with the coordinates (x_{1}, y_{1}) and (x_{2}, y_{2}).

D = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

AB = √[(-3- 3)^{2} + (5- 1)^{2}]

= =

= 2√13

BC = √[(3- 0)^{2} + (1- 3)^{2}]

=

=

CD = √[(0 + 1)^{2} + (3+ 4)^{2}]

= =

= 5√2

AD = √[(-3+ 1)^{2} + (5+ 4)^{2}]

=

=

We can observe that all sides of this quadrilateral are of different lengths.

Therefore, it can be said that it is only a general quadrilateral, and not specific such as square, rectangle, etc

(iii)Let the points (4, 5), (7, 6), (4, 3), and (1, 2) be representing the vertices A, B, C, and D of the given quadrilateral respectively

**Question 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)

**Answer:**

(i) We know that distance between the two points is given by:

d = √[(x_{1} – x_{2})^{2} + (y_{1} – y_{2})^{2}]

Distance between A(2, 3) and B(4, 1) is:

f =√ [(4 - 2)^{2} + (1 – 3)^{2}]

= √[4 + 4]= √8

= 2√2

(ii) Distance between A(-5, 7) and B(-1, 3) is:

f = √[(-5 + 1)^{2} + (7 – 3)^{2}]

= √[16 + 16] = √32

= 4√2

(iii) Distance between A(a, b) and B(-a, -b) is:

f = √[(a - (- a))

^{2}+ ( b - (-b))

^{2}]

f = √[(a+ a)^{2} + (b+ b)^{2}]

f = √[2a + 2b] = √(4a^{2} + 4b^{2})

f = 2√(a^{2} + b^{2})

**Question 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

**Answer:**

**To find: Distance between two points****Given: Points A(0, 0), B(36, 15)**

For two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}),

Distance is given by

f = [(x_{2 - }x_{1})^{2} + (y_{2 - }y_{1})^{2}]^{1/2}

Distance between (0, 0) and (36, 15) is:

f = [(36 – 0)^{2} + (15 – 0)^{2}]^{1/2}

= [1296 + 225]^{1/2} = (1521)^{1/2}

= 39

Hence Distance between points A and B is 39 unitsYes, we can find the distance between the given towns A and B. Let us take town A at origin point (0, 0)

Hence, town B will be at point (36, 15) with respect to town A

And, as calculated above, **the distance between town A and B will be 39 km**

**Question 3.**

Determine if the points (1, 5), (2, 3) and (– 2, – 11) are collinear.

**Answer:**

Let the points (1, 5), (2, 3), and (- 2,-11) be representing the vertices A, B, and C of the given triangle respectively.

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

Case 1)

Since AB + BC ≠ CA

Case 2)

Now,

BA + AC ≠ BC

Case 3)

Now,

As BC+CA≠BA

As three of the cases are not satisfied.

Hence the points are not collinear.

**Question 4.**

Check whether (5, – 2), (6, 4) and (7, – 2) are the vertices of an isosceles triangle

**Answer:**

Distance between two points A(x_{1}, y_{1}) and B(x_{2}, y_{2}) is given by

D = √(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2}

Let us assume that points (5, - 2), (6, 4), and (7, - 2) are representing the vertices A, B, and C of the given triangle respectively as shown in the figure.

AB = [(5- 6)^{2} + (-2- 4)^{2}]^{1/2}

=

=

BC = [(6- 7)^{2} + (4 + 2)^{2}]^{1/2}

=

=

CA = [(5- 7)^{2} + (-2+ 2)^{2}]^{1/2}

=

= 2

Therefore, AB = BC

As two sides are equal in length, therefore, ABC is an isosceles triangle

**Question 5.**

In a classroom, 4 friends areseated at the points A, B, C and D as shown in Fig. 7.8 Champaand Chameli walk into the classand after observing for a fewminutes Champa asks Chameli,“Don’t you think ABCD is asquare?” Chameli disagrees.Using distance formula, findwhich of them is correct

**Answer:**

It can be seen that A (3, 4), B (6, 7), C (9, 4), and D (6, 1) are the positions of 4 friends

distance between two points A(x_{1, }y_{1}) and B(x_{2}, y_{2}) is given by

D = √(x_{2} - x_{1})^{2 }+ (y_{2 }- y_{1})^{2}

Hence,

AB = [(3- 6)^{2} + (4 - 7)^{2}]^{1/2}

=

=

BC = [(6 - 9)^{2} + (7- 4)^{2}]^{1/2}

=

=

CD = [(9- 6)^{2} + (4- 1)^{2}]^{1/2}

=

=

AD = [(3 - 6)^{2} + (4 - 1)^{2}]^{1/2}

=

=

Diagonal AC = [(3 - 9)^{2} + (4 - 4)^{2}]^{1/2}

=

= 6

Diagonal BD = [(6- 6)^{2} + (7- 1)^{2}]^{1/2}

=

= 6

It can be seen that all sides of quadrilateral ABCD are of the same length and diagonals are of the same length

Therefore, ABCD is a square and hence, Champa was correct

**Question 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)

**Answer:**

**To Find: Type of quadrilateral formed**

(i) Let the points ( - 1, - 2), (1, 0), ( - 1, 2), and ( - 3, 0) be representing the vertices A, B, C, and D of the given quadrilateral respectively

The distance formula is an algebraic expression used to determine the distance between two points with the coordinates (x_{1}, y_{1}) and (x_{2}, y_{2}).

_{2}- x

_{1})

^{2}+ (y

_{2}- y

_{1})

^{2}

AB = [(-1- 1)^{2} + (-2- 0)^{2}]^{1/2}

= =

= 2√2

BC = √[(1 + 1)^{2} + (0- 2)^{2}]

= =

= 2√2

CD = √[(-1 + 3)^{2} + (2- 0)^{2}]

= =

= 2√2

AD = √[(-1+ 3)^{2} + (-2- 0)^{2}]

= =

= 2√2

Diagonal AC = √[(-1 + 1)^{2} + (-2 - 2)^{2}]

=

= 4

Diagonal BD = √[(1 + 3)^{2} + (0- 0)^{2}]

=

= 4

It is clear that all sides of this quadrilateral are of the same length and the diagonals are of the same length. Therefore, the given points are the vertices of a square

(ii)Let the points (- 3, 5), (3, 1), (0, 3), and ( - 1, - 4) be representing the vertices A, B, C, and D of the given quadrilateral respectively

The distance formula is an algebraic expression used to determine the distance between two points with the coordinates (x_{1}, y_{1}) and (x_{2}, y_{2}).

_{2}- x

_{1})

^{2}+ (y

_{2}- y

_{1})

^{2}

AB = √[(-3- 3)^{2} + (5- 1)^{2}]

= =

= 2√13

BC = √[(3- 0)^{2} + (1- 3)^{2}]

=

=

CD = √[(0 + 1)^{2} + (3+ 4)^{2}]

= =

= 5√2

AD = √[(-3+ 1)^{2} + (5+ 4)^{2}]

=

=

We can observe that all sides of this quadrilateral are of different lengths.

Therefore, it can be said that it is only a general quadrilateral, and not specific such as square, rectangle, etc

(iii)Let the points (4, 5), (7, 6), (4, 3), and (1, 2) be representing the vertices A, B, C, and D of the given quadrilateral respectively