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प्रश्न
Show that each of the triangles whose vertices are given below are isosceles :
(i) (8, 2), (5,-3) and (0,0)
(ii) (0,6), (-5, 3) and (3,1).
उत्तर
(i) Let A (8, 2) B (5, -3) and C (0, 0) be the given point
Then AB = `sqrt((8 - 5)^2 + (2 - 3)^2)`
= `sqrt(9 + 25) = sqrt(34)"units"`.
BC = `sqrt((5 - 10)^2 + (-3 - 0)^2)`
= `sqrt(25 + 9) = sqrt(34)"units"`.
AC = `sqrt((8 - 0)^2 + (2 - 0)^2)`
= `sqrt(64 + 4) = sqrt(68)."units"`.
Here AB = BC = `sqrt(34)`.
Hence, the triangle is isosceles.
(ii) The given points are A(0, 6), B(-5, 3) and C(3, 1)
Then AB = `sqrt((0 + 5)^2 + (6 - 3^2)`
= `sqrt(25 + 9) = sqrt(34)"units"`.
BC = `sqrt((-5 - 3)^2 + (3 - 1)^2)`
= `sqrt(64 + 4) = sqrt(68)"units"`.
Also AC = `sqrt((0 - 3)^2 + (6 - 1)^2)`
= `sqrt(9 + 25) = sqrt(34)"units"`.
Since AB = AC = `sqrt(34)`
Hence, the triangle is an isosceles.
Hence proved.
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Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
Each team plays with 11 players on the field during the game including the goalie. Positions you might play include -
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- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
- Goalie: As shown by player K.
Using the picture of a hockey field below, answer the questions that follow:
The coordinates of the centroid of ΔEHJ are ______.
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- 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)
- 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. - 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.
