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प्रश्न
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
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उत्तर
The distance d between two points `(x_1,_y1)` and `(x_2, y_2)` is given by the formula
`d = sqrt((x_1 - x_2)^2 + (y_1 - y_2)^2)`
In a rhombus all the sides are equal in length.
Here the four points are A (2, −1), B (3, 4), C (−2, 3) and D (−3, −2).
First let us check if all the four sides are equal.
`AB = sqrt((2 - 3)^2 + (-1 - 4)^2)`
`= sqrt((-1)^2 + (-5)^2)`
`= sqrt(1 + 25)`
`AB = sqrt26`
`BC = sqrt((3 + 2)^2 + (4 - 3)^2)`
`= sqrt((5)^2 + (1)^2)`
`= sqrt(25 + 1)`
`BC = sqrt26`
CD = sqrt((3 + 2)^2 + (4 - 3)^2)
`= sqrt((-5)^2 + (-1)^2)`
`=sqrt(25 + 1)`
`CD = sqrt26`
`AD = sqrt((2 + 3)^2 + (-1 + 2)^2)`
`= sqrt((5)^2 + (1)^2)`
`= sqrt(25 + 1)`
`AD = sqrt26`
Here, we see that all the sides are equal, so it has to be a rhombus.
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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 -
- Forward: As shown by players A, B, C and D.
- 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 ______.
