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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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संबंधित प्रश्न
If A(5, 2), B(2, −2) and C(−2, t) are the vertices of a right angled triangle with ∠B = 90°, then find the value of t.
If the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), prove that bx = ay
If P (2, – 1), Q(3, 4), R(–2, 3) and S(–3, –2) be four points in a plane, show that PQRS is a rhombus but not a square. Find the area of the rhombus
Prove that the points A(1, 7), B (4, 2), C(−1, −1) D (−4, 4) are the vertices of a square.
Find the distance between the points
A(-6,-4) and B(9,-12)
P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.
Prove that the points (0 , 2) , (1 , 1) , (4 , 4) and (3 , 5) are the vertices of a rectangle.
Find the distance between the origin and the point:
(-8, 6)
Find the distance between the origin and the point:
(8, −15)
A point P lies on the x-axis and another point Q lies on the y-axis.
If the abscissa of point P is -12 and the ordinate of point Q is -16; calculate the length of line segment PQ.
Show that the points P (0, 5), Q (5, 10) and R (6, 3) are the vertices of an isosceles triangle.
Prove that the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS.
Show that the points A (5, 6), B (1, 5), C (2, 1) and D (6, 2) are the vertices of a square ABCD.
Find the point on y-axis whose distances from the points A (6, 7) and B (4, -3) are in the ratio 1: 2.
Find the distance of the following points from origin.
(5, 6)
Find distance between point A(–1, 1) and point B(5, –7):
Solution: Suppose A(x1, y1) and B(x2, y2)
x1 = –1, y1 = 1 and x2 = 5, y2 = – 7
Using distance formula,
d(A, B) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
∴ d(A, B) = `sqrt(square +[(-7) + square]^2`
∴ d(A, B) = `sqrt(square)`
∴ d(A, B) = `square`
The distance between the point P(1, 4) and Q(4, 0) is ______.
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 ______.
The distance between the points (0, 5) and (–3, 1) is ______.
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
