**Q) **Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane.

(i) At an instance, the midfielders and forward formed a parallelogram. Find the position of the central midfielder (D) if the position of other players who formed the parallelogram are:- A(1,2), B(4,3) and C(6,6)

(ii) Check if the Goal keeper G(-3,5), Sweeper H(3,1) and Wing-back K(0,3) fall on a same straight line.

[or]

Check if the Full-back J(5,-3) and centre-back I(-4,6) are equidistant from forward C(0,1) and if C is the mid-point of IJ.

(iii) If Defensive midfielder A(1,4), Attacking midfielder B(2,-3) and Striker E(a,b) lie on the same straight line and B is equidistant from A and E, find the position of E.

**Ans:**

**(i) Position of the central midfielder (D):**

Let’s consider position coordinates of point D are (a, b)

Since, it is given ABCD is parallelogram, therefore its both diagonals will bisect each other.

Hence, mid point of AC = mid point of BD

4 + a = 7 or a = 3

and b + 3 = 8 or b = 5

**Therefore, the position coordinates of point D are (3, 5).**

**(ii) Point GHK fall on same straight line or not:**

If three points are on same straight line, then the line length will be exactly equal to sum of its segments.

Let’s calculate length of each of these lines by given coordinates of all three points:

GH:

GK:

HK:

We can clearly see that GH = GK + HK, **therefore, points GHK are on a same straight line.**

**OR**

**A) Points J and I are equidistant from C:**

If points, J & I are equidistant from point C, then JC = IC

Let’s calculate length of each of these lines by given coordinates:

JC:

IC:

Since both lengths are equal, **therefore points J & I are equidistant from point C.**

**B) If C is the mid-point of IJ:**

Let’s calculate coordinates of Mid point of IJ:

Since coordinates of C are given as (0,1); **therefore C is not the midpoint of IJ.**

(iii) If Defensive midfielder A(1,4), Attacking midfielder B(2,-3) and Striker E(a,b) lie on the same straight line and B is equidistant from A and E, find the position of E.

Given that points, A, B and E are in the straight line and B is equidistant from A and E, it clearly means that B is the mid-point of AE i.e. AB = BE.

By B being mid-point of AE, we get:

= (2, -3)

From X-axis coordinate, we get: = 2

1 + a = 4 or **a = 3**

From Y-axis coordinate, we get: = – 3

4 + b = – 6 or b** = -10**

**Therefore, If Defensive midfielder A(1, 4), Attacking midfielder B(2, -3) and Striker E(a, b) lie on the same straight line and B is equidistant from A and E, then position coordinates of E are (3, -10).**