Class 10 RD Sharma Solutions – Chapter 14 Coordinate Geometry – Exercise 14.3 | Set 3

Question 41. Three consecutive vertices of a parallelogram are (-2, -1), (1, 0) and (4, 3). Find the fourth vertex.

Solution:

Let the coordinates of three vertices are A (-2, -1), B (1, 0), and C (4, 3)

And let the diagonals AC and BD bisect each other at O

As O is the mid-point of AC

Therefore, 

Vertices of O will be

Assume coordinates of the forth vertex D be (x, y)

As O is the mid-point of BD

Thus, coordinates of O will be 

Therefore(1 + x)/2 = 1

1 + x = 2

x = 1

and

y/2 = 1

y = 2

Hence, the co-ordinates of D will be (1, 2).

Question 42. The points (3, -4) and (-6, 2) are the extremities of a diagonal of a parallelogram. If the third vertex is (-1, -3). Find the co-ordinates of the fourth vertex.

Solution:

Assume the edges of a diagonal AC of a parallelogram ABCD are A (3, -4) and C (-6, 2)

Consider AC and BD bisect each other at O.

Therefore,

Mid-point of AC will be

Assume the fourth vertex of the parallelogram be (x, y)

Therefore,

Mid-point of BD will be 

-1 + x = -3 

x = -2

and

(-3 + y)/2 = -1

-3 + y = -2 

y = -2 + 3 = 1

Hence, the coordinates of D are (-2, 1).

Question 43. If the co-ordinates of the mid-points of the sides of a triangle are (1, 1), (2, -3), and (3, 4), find the vertices of the triangle.

Solution:

Assume A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) be the vertices of the ∆ABC

D, E and F are the mid-points of BC, CA and AB respectively such that their co-ordinates are D (1, 1), E (2, -3) and F (3, 4)

D is mid-point of BC

Therefore,

x₂ + x₃ = 2

and

y₂ + y₃ = 2

Similarly, E is the mid-point of AC

Therefore,

x₃ + x₁ = 4

and

y₃ + y₁ = -6

And

F is the mid-point of AB

Therefore,

x₂ + x₁ = 6

and

y₂ + y₁ = 8

Now,

x₁ + x₂ = 6  …………..(i)

x₂ + x₃ = 2  ……………(ii)

x₃ + x₁ = 4  …………….(iii)

On adding we will get

2(x₁ + x₂ + x₃) = 12

x₁ + x₂ + x₃ = 6  …………(iv)

On subtracting (ii), (iii) and (i) from (iv), we get

 x₁ = 4, x₂ = 2, x₃ = 0

Similarly

y₁ + y₂ = 8  ……….(v)

y₂ + y₃ = 2  ……….(vi)

y₃ + y₁ = -6  ………(vii)

On adding we will get 

2(y₁ + y₂ + y₃) = 4

y₁ + y₂ + y₃ = 2  ………(viii)

On subtracting (vi), (vii) and (v) from (viii), we get

y₁ = 0

y₂ = 8

y₃ = -6

Hence, the vertices of ∆ABC are A (4, 0), B(2, 8), C(0, -6)

Question 44. Determine the ratio in which the straight line x – y – 2 = 0 divides the line segment joining (3, -1) and (8, 9).

Solution:

Assume the straight line x – y – 2 = 0 divides the line segment joining the points (3, -1), (8, 9) in the ratio m : n

Co-ordinates of the point will be

and

This point (x, y) lies on the line on the line x – y – 2 = 0

⇒ (8m + 3n) – (9m – n) – 2(m + n) = 0

⇒ 8m + 3n – 9m + n – 2m – 2n = 0

⇒ -3m + 2n = 0

⇒ 2n = 3m

Hence, the ratio = 2 : 3 internally

Question 45. Three vertices of a parallelogram are (a + b, a – b), (2 a + b, 2a – b), (a – b, a + b). Find the fourth vertex. 

Solution:

In parallelogram ABCD co-ordinates are of A (a + b, a – b), B (2a + b, 2a – b), C (a – b, a + b)

Assume coordinates of D be (x, y)

Join diagonal AC and BD

Which bisect each other at O

O is the mid-point of AC as well as BD

If O is the mid-point of AC, Then its coordinates will be

and 

If O is mid-point of BD, then coordinates will be

x + 2a + b = 2a

x = 2a – 2a – b = -b

and

y + 2a – b = 2a

y = 2a – 2a + b = b

Hence, the coordinates of D will be (-b, b).

Question 46. If two vertices of a parallelogram are (3, 2), (-1, 0) and the diagonals cut at (2, -5), find the other vertices of the parallelogram.

Solution:

Two vertices of a parallelogram ABCD are A (3,2), and B (-1, 0) and its diagonals bisect each other at O (2, -5)

Assume the coordinates of C be (x₁, y₁) and of D be (x₂, y₂)

If O is the mid-point of AC, then

3 + x₁ = 4

x₁ = 4 – 3 = 1

and 

and

2 + y₁ = -10

y₁ = -10 – 2 = -12

Therefore, the coordinates of C will be (1, -12)

Again if O is mid-point of BD then

-1 + x₂ = 4

x₂ = 4 + 1 = 5

and

y₂ = -10

Hence, the coordinates of D will be (5, -10).

Question 47. If the coordinates of the mid-points of the sides of a triangle ar6 (3, 4), (4, 6), and (5, 7), find its vertices. 

Solution:

The coordinates of the mid-points of the sides BC, CA and AB are D (3, 4), E (4, 6) and F (5, 7) of the ∆ABC.

Assume the coordinates of the vertices of the triangle be A (x,₁ y₁), B (x₂, y₂), and C (x₃, y₃).

Now the coordinates of D will be

x₂ + x₃ = 6

and

y₂ + y₃ = 4

y₂ + y₃ = 8

Similarly, the coordinates of E will be

 x₃ + x₁ = 8

and

and

y₁ + y₂ = 14

Now,

x₂ + x₃ = 6  …….(i)

x₃ + x₁ = 8  ……..(ii)

x₁ + x₂ = 10  ……..(iii)

On adding we will get

2(x₁ + x₂ + x₃) = 24

⇒ x₁ + x₂ + x₃ = 24/2 = 12  ………..(iv)

On subtracting each from (iv), 

We will get

x₁ = 6, x₂ = 4 and x₃ = 2

Similarly,

y₂ + y₃ = 8   ……..(v)

y₃ + y₁ = 12  ………(vi)

y₁ + y₂ = 14   ……..(vii)

On Adding, we will get

2(y₁ + y₂ + y₃) = 34

⇒ y₁ + y₂ + y₃ = 34/2  = 17  ……..(viii)

On subtracting each from (viii),

We will get

y₁ = 9

y₂ = 5

y₃ = 3

Hence, the coordinates will be of A (6, 9), B (4, 5) and C (2, 3)

Question 48. The line segment joining the points P (3, 3) and Q (6, -6) is trisected at the points A and B such that A is nearer to P. If A also lies on the line given by 2x+ y + k =0, find the value of k.

Solution:

Two points A and B trisect the line segment joining the points P (3, 3) and Q (6, -6) and A is nearer to P and A lies also on the line 2x + y + k = 0

Now,

A divides the line segment PQ in the ratio of 1 : 2

i.e.,

PA = AQ = 1 : 2

Assume coordinates of A be (x, y), then

and

Therefore,

Coordinates of A are (4, 0)

As A lies on the line 2x + y + k = 0

Hence,

It will satisfy it

2 × 4 + 0 + k = 0

8 + k = 0

k = -8

Question 49. If three consecutive vertices of a parallelogram are (1, -2), (3, 6), and (5, 10), find its fourth vertex.

Solution:

A (1, -2), B (3, 6) and C (5, 10) are the three consecutive vertices of the parallelogram ABCD

Assume (x, y) be its fourth vertex

AC and BD are its diagonals which bisect each other at O

As O is the mid-point of AC

Therefore,

Coordinates of O will be

On comparing, 

3 + x = 3

3 + x = 6

x = 3

and

(6 + y)/2 = 4

6 + y = 8

y = 2

Hence, the coordinates of fourth vertex D are (3, 2).

Question 50. If the points A (a, -11), B (5, b), C (2, 15), and D (1, 1) are the vertices of a parallelogram ABCD, find the values of a and b.

Solution:

A (a, -11), B (5, b), C (2, 15) and D (1, 1) are the vertices of a parallelogram ABCD

Diagonals AC and BD bisect each other at O

As O is the mid-point of AC

Therefore,

Coordinates of O will be

Similarly, O is the mid-point of BD also

Therefore,

Coordinates of O will be

(2 + a)/2 = 3

b + 1 = 4

b = 3

Hence, 

a = 4 

b = 3

Question 51. If the co-ordinates of the mid-points of the sides of a triangle be (3, -2), (-3, 1), and (4, -3), then find the coordinates of its vertices.

Solution:

In a ∆ABC,

D, E and F are the mid-points of the sides BC, CA and AB respectively and co-ordinates of D, E and F are (3, -2), (-3, 1) and (4, -3) respectively.

Assume the coordinates of A are (x₁, y₁), of B are (x₂, y₂) and of C are (x₃, y₃)

Now,

As D is the mid-point BC

Therefore,

and 

As E is the mid-point of CA

Therefore,

and

and F is the mid-point of AB

Therefore,

and

Now,

x₂ + x₃ = 6  ……..(i)

x₃ + x₁ = -6  ……..(ii)

x₁ + x₂ = 8   ………(iii)

On adding we get

2(x₁ + x₂ + x₃) = 8

⇒ x₁ + x₂ + x₃ = 4   …….(iv)

On subtracting from (iv) we get

x₁ = -2, 

x₂ = 10,

x₃ = -4

and

y₂ + y₃ = -4   …….(v)

y₃ + y₁ = 2   …….(vi)

y₁ + y₂ = -6  ………(vii)

On Adding, We will get

2(y₁ + y₂ + y₃) = -8

⇒ y₁ + y₂ + y₃ = -4   ……..(viii)

On subtracting from (viii) we get

y₁ = 0,

y₂ = -6,

y₃ = 2

Hence, the coordinates of A are (-2, 0) of B are (10, -6) and of C (-4, 2).

Question 52. The line segment joining the points (3, -4) and (1, 2) is trisected at the points P and Q. If the co-ordinates of P and Q are (p, -2) and (53 , q) respectively, find the values of p and q. 

Solution:

Assume line AB whose ends points are A (3, -4) and B (1, 2)

Coordinates of P and Q which trisect AB are P (p, -2) and Q 

Now,

As P divides AB in the ratio 1 : 2

Therefore,

Coordinates of P will be

Similarly, Q divides AB in the ratio 2 : 1

Hence,

p = 7/3, q = 0

Question 53. The line joining the points(2, 1), (5, -8) is trisected at the points P and Q. If point P lies on the line 2x – y + k = 0, find the value of k.

Solution:

Points A (2, 1), and B (5, -8) are the ends points of the line segment AB

Points P and Q trisect it and P lies on the line 2x – y + k = 0

As P divides AB in the ratio of 1 : 2

Therefore,

Coordinates of P will be

Therefore,

Coordinates of P are (3, -2)

As it lies on 2x – y + k = 0

Therefore, 

It will satisfy it

2 × 3 – (-2) + k = 0

⇒ 6 + 2 + k = 0

⇒ 8 + k = 0

⇒ k = -8

Question 54. A (4, 2), B (6, 5), and C (1, 4) are the vertices of ∆ABC,

(i) The median from A meets BC in D. Find the coordinates of the point D.

(ii) Find the coordinates of point P on AD such that AP : PD = 2 : 1.

(iii) Find the coordinates of the points Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.

(iv) What do you observe? 

Solution:

In ∆ABC, co-ordinates of A (4, 2) of (6, 5) and of (1, 4) and AD is BE and CF are the medians

 such that D, E and F are the mid-points of the sides BC, CA and AB respectively

P is a point on AD such that AP : PD = 2 : 1

(i) Now coordinates of D will be

(ii) As P divides AD in the ratio of 2 : 3

Therefore,

Coordinates of P will be

Thus,

Coordinates of P are 

(iii) As E and F are the mid-point if CA and AB respectively

Coordinates of E will be 

and of F will be

As Q and R divides BE and CF in such a way that BQ : QE = 2 : 1 and CR : RF = 2 : 1

Therefore,

Coordinates of Q will be

and 

i.e., Q₁ is (11/3, 11/3)

and similarly the coordinates of R will be

and

R is (11/3, 11/3)

(iv) We can see that coordinates of P, Q and R are same 

i.e., P, Q and R coincides each other. 

Medians of the sides of a triangle pass through the same point which is called the centroid of the triangle.

Question 55. If the points A (6, 1), B (8, 2), C (9, 4), and D (k, p) are the vertices of a parallelogram taken in order, then find the values of k and p.

Solution:

The diagonals of a parallelogram bisect each other

O is the mid-point of AC and also of BD

O is the mid-point of AC

Therefore,

Coordinates of O will be

As O is also the mid-point of BD

Therefore,

and

⇒ 8 + k = 15

⇒ k = 15 – 8 = 7

and

⇒ 2 + p = 5

⇒ p = 5 – 2 = 3

Hence, 

k = 7, p = 3

Question 56. A point P divides the line segment joining the points A (3, -5) and B (-4, 8) such that . If P lies on the line x + y = 0, then find the value of k. 

Solution:

Point P divides the line segment by joining the points A (3, -5) and B (-4, 8)

Such that 

⇒ AP : PB = k : 1

Assume coordinates of P be (x, y), then

and

As x + y = 0

Therefore,

4k – 2 = 0

4k = 2

k = 2/4

k = 1/2 

Question 57. The mid-point P of the line segment joining the points A (-10, 4) and B (-2, 0) lies on the line segment joining the points C (-9, -4) and D (-4, y). Find the ratio in which P divides CD. Also, find the value of y. 

Solution:

P is the mid-point of line segment joining the points A (-10, 4) and B (-2, 0)

Coordinates of P will be

P lies on CD also,

Let P divides C (9, 4) and D (-4, y) in the ratio m₁ : m₂

Therefore,

⇒ -6m₁ – 6m₂ = -4m₁ – 9m₂

⇒ -6m₁ + 4m₁ = -9m₂ + 6m₂

⇒ -2m₁ = -3m₂

⇒ 

Therefore,

m₁ : m₂ = 3 : 2

and

⇒ 10 = 3y – 8x 

⇒ 3y = 10 + 8 = 18

⇒ y = 18/3 = 6

⇒ y = 6

Question 58. If the point C (-1, 2) divides internally the line segment joining the points A (2, 5) and B (x, y) in the ratio 3 : 4, find the value of x² + y². 

Solution:

As we know that if a point (x, y) divides the line segment joining 

the points (x₁, y₁) and (x,₂ y₂) in the ration m : n, then

and

Now here, C (-1, 2) divides the line segment joining A (2, 5) and B (x, y) in the ratio of 3 : 4

Now, 

and

⇒ 3x + 8 = -7

and

⇒ 20 – 3y = 14

⇒ x = -5

and

⇒ y = 2

Now,

x² + y² = (-5)² + (2)²

x² + y² = 25 + 4 = 29

Question 59. ABCD is a parallelogram with vertices A (x₁, y₁), B (x₂, y₂), and C (x₃, y₃). Find the coordinates of the fourth vertex D in terms of x₁, x₂, x₃, y₁, y₂, and y₃ 

Solution:

Assume the coordinates of D be (x, y). We know that diagonals of a parallelogram bisect each other.

Hence, 

Mid-point of AC = Mid-point of BD 

i.e., x₁ + x₃ = x₂ + x and y₁ + y₃ = y₂ + y

i.e, x₁ + x₃ – x₂ = x and y₁ + y₃ + y₂ = y

Hence, the coordinates of D are (x₁ + x₃ – x₂, y₁ + y₃ + y₂)

Question 60. The points A (x₁, y₁), B (x₂, y₂), and C (x₃, y₃) are 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 points of coordinates Q and R on medians BE and CF respectively such that BQ : QE = 2 : 1 and CR : RF = 2 : 1.

(iv) What are the coordinates of the centroid of the triangle ABC? 

Solution:

Given: The points A (x₁, y₁), B (x₂, y₂) and C (x₃, y₃) are the vertices of ∆ABC.

(i) We know that, the median bisect the line segment into two equal parts i.e., here D is the mid-point of BC.

Therefore,

Coordinates of mid-point of BC = 

D = 

(ii) Assume the coordinates of a point P be (x, y)

Given that, the point P (x, y), divide the line joining A (x_1, y₁) and 

D  in the ration of 2 : 1,

Then the coordinates of P

Using internal section formula, we get

Therefore,

Required coordinates of points P = 

(iii) Assume the coordinates of a point Q be (p, q)

Given: The point Q (p, q) divide the line joining B(x_2, y₂) and E 

In the ratio of 2 : 1

Then the coordinates of Q

[Since, BE is the median of side CA. So, BE divides AC into two equal parts]

Therefore,

Mid-point of AC = Coordinates of E

⇒ E = 

So, the required coordinate of point Q = 

Now, let us considered that the coordinates of a point E be (α, β). 

It is given that, the point R (α, β), divide the line joining C (x₃, y₃) and 

F  in the ration 2 : 1, 

So, the coordinates of R be

Here, CF is the median of side AB, hence CF divides AB into two equal parts

Hence, 

Mid-point of AB = Coordinates of CF

⇒ F 

So, required coordinate of point R 

(iv) Coordinate of the centroid of the ∆ABC = (sum of abscissa of all vertices/3, sum of all vertices/2)