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प्रश्न
Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
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उत्तर
The distance d between two points `(x_1,y_1) and (x_2, y_2)` is given by the formula
`d = sqrt((x_1 - x_2)^2 +(y_1 - y_2)^2)`
In a parallelogram the opposite sides are equal in length.
Here the four points are A(1, −2), B(3, 6), C(5, 10) and D(3, 2).
Let us check the length of the opposite sides of the quadrilateral that is formed by these points.
`AB = sqrt((1 - 3)^2 + (2 - 6))`
`= sqrt((-2)^2 + (-8)^2)`
`= sqrt(4 + 64)`
`AB = sqrt(68)`
`CD = sqrt((5 - 3)^2 + (10 - 2)^2)`
`= sqrt((2)^2 + (8)^2)`
`= sqrt(4 + 64)`
`CD = sqrt(68)`
We have one pair of opposite sides equal.
Now, let us check the other pair of opposite sides.
`BC = sqrt((3 - 5)^2 + (6 - 10)^2)`
`= sqrt((-2)^2 + (-4)^2)`
`=sqrt(4 + 16)`
`AD = sqrt20`
The other pair of opposite sides is also equal. So, the quadrilateral formed by these four points is definitely a parallelogram.
Hence we have proved that the quadrilateral formed by the given four points is a parallelogram
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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:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
Point P(0, 2) is the point of intersection of y-axis and perpendicular bisector of line segment joining the points A(–1, 1) and B(3, 3).
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
