Question

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.

1. 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)
2. 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.
3. 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.

Answer 1

i. Let D be (a, b), then

Mid point of AC = Midpoint of BD

`((1 + 6)/2, (2 + 6)/2) = ((4 + a)/2, (3 + b)/2)`

4 + a = 7

a = 3

3 + b = 8

b = 5

Central midfielder is at (3, 5)

ii. GH = `sqrt((-3 - 3)^2 + (5 - 1)^2`

= `sqrt(36 + 16)`

= `sqrt(52)`

= `2sqrt(13)`

GK = `sqrt((0 + 3)^2 + (3 - 5)^2`

= `sqrt(9 + 4)`

= `sqrt(13)`

HK = `sqrt((3 - 0)^2 + (1 - 3)^2`

= `sqrt(9 + 4)`

= `sqrt(13)`

GK + HK = GH `\implies` G,H and K lie on a same straight line

[or]

CJ = `sqrt((0 - 5)^2 + (1 + 3)^2`

= `sqrt(25 + 16)`

= `sqrt(41)`

CI = `sqrt((0 + 4)^2 + (1 - 6)^2`

= `sqrt(16 + 25)`

= `sqrt(41)`

Full-back J(5, –3) and centre-back I(–4, 6) are equidistant from forward C(0, 1)

Mid-point of IJ = `((5 - 4)/2, (-3 + 6)/2) = (1/2, 3/2)`

C is NOT the mid-point of IJ

iii. A, B and E lie on the same straight line and B is equidistant from A and E

⇒ B is the mid-point of AE

`((1 + a)/2, (4 + b)/2)` = (2, – 3)

1 + a = 4 ; a = 3.

4 + b = -6; b = –10 E is (3, –10)