Class 10 - Chapter 7: Coordinate Geometry

Exercise 7.2

Q1. Find the coordinates of the point which divides the join of (−1, 7) and (4, −3) in the ratio 2 : 3.

Let A(−1, 7) and B(4, −3).

The point P divides AB internally in the ratio 2 : 3.

Using the Section Formula:

P = [ (mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n) ]

Here,

x₁ = −1, y₁ = 7

x₂ = 4, y₂ = −3

m = 2, n = 3

x-coordinate:

= [2(4) + 3(−1)]/(2 + 3)

= (8 − 3)/5

= 1

y-coordinate:

= [2(−3) + 3(7)]/(2 + 3)

= (−6 + 21)/5

= 3

Required point = (1, 3)

Q2. Find the coordinates of the points of trisection of the line segment joining (4, −1) and (−2, −3).

Let A(4, −1) and B(−2, −3).

The points of trisection divide the line segment in the ratios 1 : 2 and 2 : 1.

First point P (ratio 1 : 2)

P = [ (1×(−2) + 2×4)/(1 + 2), (1×(−3) + 2×(−1))/(1 + 2) ]

= (6/3, −5/3)

= (2, −5/3)

Second point Q (ratio 2 : 1)

Q = [ (2×(−2) + 1×4)/(2 + 1), (2×(−3) + 1×(−1))/(2 + 1) ]

= (0/3, −7/3)

= (0, −7/3)

The points of trisection are (2, −5/3) and (0, −7/3).

Q3. To conduct Sports Day activities, in your rectangular shaped school ground ABCD, lines have been drawn with chalk powder at a distance of 1m each. 100 flower pots have been placed at a distance of 1m from each other along AD, as shown in Fig. Niharika runs 1/4 th the distance AD on the 2nd line and posts a green flag. Preet runs 1/5 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 joining the two flags, where should she post her flag?

Source- NCERT

From the figure,

AD = 100 m

The lines are 1 m apart.

Niharika runs on the 2nd line and posts a green flag at a distance 1/4 of AD from A.

Green flag coordinates = (2, 25)

Preet runs on the 8th line and posts a red flag at a distance 1/5 of AD from A.

Red flag coordinates = (8, 20)

Distance between the two flags

d = √[(8 − 2)² + (20 − 25)²]

= √[36 + 25]

= √61 m

≈ 7.81 m

Position of Rashmi's Blue Flag

The blue flag is to be posted at the midpoint of the line segment joining the two flags.

Midpoint Formula:

M = [ (x₁ + x₂)/2 , (y₁ + y₂)/2 ]

= [ (2 + 8)/2 , (25 + 20)/2 ]

= (5, 45/2)

= (5, 22.5)

Distance between the flags = √61 m.

Rashmi should post the blue flag at (5, 22.5).

Q4. Find the ratio in which the line segment joining the points (−3, 10) and (6, −8) is divided by (−1, 6).

Let A(−3, 10), B(6, −8) and P(−1, 6).

Suppose P divides AB in the ratio m : n.

Using Section Formula:

−1 = [6m − 3n]/(m + n)

⇒ −m − n = 6m − 3n

⇒ 2n = 7m

⇒ m : n = 2 : 7

Verification using y-coordinate:

6 = [−8m + 10n]/(m + n)

⇒ 6m + 6n = −8m + 10n

⇒ 14m = 4n

⇒ m : n = 2 : 7

Hence verified.

The point (−1, 6) divides the line segment internally in the ratio 2 : 7.

Q5. 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.

Let P be the point where AB intersects the x-axis.

Since P lies on the x-axis, its y-coordinate is 0.

Suppose P divides AB in the ratio m : n.

Using the Section Formula:

0 = [m(5) + n(−5)]/(m + n)

5m − 5n = 0

m = n

Therefore,

m : n = 1 : 1

Now find the coordinates of P:

x = [m(−4) + n(1)]/(m + n)

= (−4 + 1)/2

= −3/2

y = 0

Ratio = 1 : 1

Point of division = (−3/2, 0)

Q6. If (1, 2), (4, y), (x, 6) and (3, 5) are the vertices of a parallelogram taken in order, find x and y.

Let

A(1, 2), B(4, y), C(x, 6), D(3, 5)

In a parallelogram, diagonals bisect each other.

Therefore, midpoint of AC = midpoint of BD.

Midpoint of AC

= ((1 + x)/2, (2 + 6)/2)

= ((1 + x)/2, 4)

Midpoint of BD

= ((4 + 3)/2, (y + 5)/2)

= (7/2, (y + 5)/2)

Equating x-coordinates:

(1 + x)/2 = 7/2

1 + x = 7

x = 6

Equating y-coordinates:

4 = (y + 5)/2

8 = y + 5

y = 3

x = 6 and y = 3

Q7. Find the coordinates of a point A, where AB is the diameter of a circle whose centre is (2, −3) and B is (1, 4).

The centre of a circle is the midpoint of its diameter.

Let A = (x, y)

B = (1, 4)

Centre = (2, −3)

Using the midpoint formula:

((x + 1)/2, (y + 4)/2) = (2, −3)

(x + 1)/2 = 2

x + 1 = 4

x = 3

(y + 4)/2 = −3

y + 4 = −6

y = −10

Coordinates of A = (3, −10)

Q8. If A and B are (−2, −2) and (2, −4), respectively, find the coordinates of P such that

AP = (3/7) AB and P lies on the line segment AB.

Since

AP = (3/7)AB

PB = (4/7)AB

Therefore,

AP : PB = 3 : 4

Using the Section Formula:

P = [(3×2 + 4×(−2))/(3+4), (3×(−4) + 4×(−2))/(3+4)]

= [(-2)/7, (-20)/7]

P = (−2/7, −20/7)

Q9. Find the coordinates of the points which divide the line segment joining A(−2, 2) and B(2, 8) into four equal parts.

To divide AB into four equal parts, three points are required.

These points divide AB in the ratios:

1 : 3, 2 : 2 and 3 : 1

First Point P (1 : 3)

P = [(1×2 + 3×(−2))/4, (1×8 + 3×2)/4]

= (−1, 7/2)

Second Point Q (2 : 2)

Q = [(2×2 + 2×(−2))/4, (2×8 + 2×2)/4]

= (0, 5)

Third Point R (3 : 1)

R = [(3×2 + 1×(−2))/4, (3×8 + 1×2)/4]

= (1, 13/2)

The required points are:

(−1, 7/2), (0, 5), (1, 13/2)

Q10. Find the area of a rhombus if its vertices are (3, 0), (4, 5), (−1, 4) and (−2, −1) taken in order.

Let the vertices be:

A(3, 0), B(4, 5), C(−1, 4), D(−2, −1)

Diagonal AC:

AC = √[(−1 − 3)² + (4 − 0)²]

= √(16 + 16)

= √32

= 4√2

Diagonal BD:

BD = √[(−2 − 4)² + (−1 − 5)²]

= √(36 + 36)

= √72

= 6√2

Area of Rhombus

= (1/2) × (Diagonal 1) × (Diagonal 2)

= (1/2) × (4√2) × (6√2)

= (1/2) × 24 × 2

= 24 square units

Area of the rhombus = 24 square units.