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प्रश्न
Name the type of quadrilateral formed, if any, by the following points, and give reasons for your answer:
(- 1, - 2), (1, 0), (- 1, 2), (- 3, 0)
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उत्तर
Let the points (−1, −2), (1, 0), (−1, 2), and (−3, 0) represent the vertices A, B, C, and D of the given quadrilateral, respectively.
AB = `sqrt((-1-1)^2+(-2-0)^2)`
= `sqrt((-2)^2+(-2)^2)`
= `sqrt(4+4)`
= `sqrt8`
=`2sqrt2`
BC = `sqrt((1-(-1))^2+(0-2)^2)`
= `sqrt((2)^2+(-2)^2)`
= `sqrt(4+4)`
= `sqrt8`
= `2sqrt2`
CB = `sqrt((-1-(-3))^2+(2-0)^2) `
= `sqrt((2)^2+(2)^2)`
= `sqrt(4+4)`
= `sqrt8 `
= `2sqrt2`
AD = `sqrt((-1-(3))^2 + (-2-0)^2)`
= `sqrt((2)^2+(-2)^2)`
= `sqrt(4+4)`
= `sqrt8`
= `2sqrt2`
Diagonal AC = `sqrt((-1-(-1))^2+(-2-2)^2)`
= `sqrt(0^2+(-4)^2)`
= `sqrt(16) `
= 4
Diagonal BD = `sqrt((1-(-3))^2+(0-0)^2)`
= `sqrt((4)^2+0^2)`
= `sqrt16 `
= 4
It can be observed that all sides of this quadrilateral are of the same length and also, the diagonals are of the same length. Therefore, the given points are the vertices of a square.
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संबंधित प्रश्न
If the point (x, y) is equidistant from the points (a + b, b – a) and (a – b, a + b), prove that bx = ay.
Find the coordinates of the centre of the circle passing through the points (0, 0), (–2, 1) and (–3, 2). Also, find its radius.
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
If A (-1, 3), B (1, -1) and C (5, 1) are the vertices of a triangle ABC, find the length of the median through A.
Find the distance of the following points from the origin:
(iii) C (-4,-6)
Using the distance formula, show that the given points are collinear:
(-1, -1), (2, 3) and (8, 11)
Find the distances between the following point.
R(–3a, a), S(a, –2a)
Find the distance between the following pairs of point in the coordinate plane :
(4 , 1) and (-4 , 5)
Find the distance between the following point :
(p+q,p-q) and (p-q, p-q)
Find the coordinate of O , the centre of a circle passing through P (3 , 0), Q (2 , `sqrt 5`) and R (`-2 sqrt 2` , -1). Also find its radius.
In what ratio does the point P(−4, y) divides the line segment joining the points A(−6, 10) and B(3, −8)? Hence find the value of y.
A point P (2, -1) is equidistant from the points (a, 7) and (-3, a). Find a.
The points A (3, 0), B (a, -2) and C (4, -1) are the vertices of triangle ABC right angled at vertex A. Find the value of a.
Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
If the point (x, y) is at equidistant from the point (a + b, b – a) and (a-b, a + b). Prove that ay = bx.
Show that the point (0, 9) is equidistant from the points (– 4, 1) and (4, 1)
The point which divides the lines segment joining the points (7, -6) and (3, 4) in ratio 1 : 2 internally lies in the ______.
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 of the point (5, 0) from the origin is ______.
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Tharunya was thrilled to know that the football tournament is fixed with a monthly timeframe from 20th July to 20th August 2023 and for the first time in the FIFA Women’s World Cup’s history, two nations host in 10 venues. Her father felt that the game can be better understood if the position of players is represented as points on a coordinate plane. |
- 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.
