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प्रश्न
Show that the points A (5, 6), B (1, 5), C (2, 1) and D (6, 2) are the vertices of a square ABCD.
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उत्तर
We use the distance formula = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)` to find the length of each side.
AB = `sqrt((1 - 5)^2 + (5 - 6)^2)`
= `sqrt((-4)^2 + (-1)^2)`
= `sqrt(16 +1)`
= `sqrt(17)`
BC = `sqrt((2 - 1)^2 + (1 - 5)^2)`
= `sqrt((1)^2 + (-4)^2)`
= `sqrt(1+16)`
= `sqrt(17)`
CD = `sqrt((6 - 2)^2 + (2 - 1)^2)`
= `sqrt((4)^2 + (1)^2)`
= `sqrt(16 + 1)`
= `sqrt(17)`
DA = `sqrt((5 - 6)^2 + (6 - 2)^2)`
= `sqrt((-1)^2 + (4)^2)`
= `sqrt(1+16)`
= `sqrt(17)`
To prove it is a square, the diagonals must also be equal in length.
AC = `sqrt((2 - 5)^2 + (1 - 6)^2)`
= `sqrt((-3)^2 + (-5)^2)`
= `sqrt(9+25)`
= `sqrt(34)`
BD = `sqrt((6 - 1)^2 + (2 - 5)^2)`
= `sqrt((5)^2 + (-3)^2)`
= `sqrt(25 + 9)`
= `sqrt(34)`
The diagonals have the same length, `sqrt(34)` units.
Since, AB = BC = CD = DA and AC = BD,
A, B, C and D are the vertices of a square.
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संबंधित प्रश्न
If the point P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b). Prove that bx = ay.
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Given a line segment AB joining the points A(–4, 6) and B(8, –3). Find
1) The ratio in which AB is divided by y-axis.
2) Find the coordinates of the point of intersection.
3) The length of AB.
Find the centre of the circle passing through (6, -6), (3, -7) and (3, 3)
Using the distance formula, show that the given points are collinear:
(-2, 5), (0,1) and (2, -3)
Find the distance of the following point from the origin :
(8 , 15)
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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 point on y axis equidistant from B and C is ______.
What type of a quadrilateral do the points A(2, –2), B(7, 3), C(11, –1) and D(6, –6) taken in that order, form?
Find the value of a, if the distance between the points A(–3, –14) and B(a, –5) is 9 units.
