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प्रश्न
In a classroom, 4 friends are seated at the points A, B, C and D as shown in the following figure. Champa and Chameli walk into the class and after observing for a few minutes, Champa asks Chameli, “Don’t you think ABCD is a square?” Chameli disagrees.
Using distance formula, find which of them is correct.
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उत्तर
It can be observed that A (3, 4), B (6, 7), C (9, 4), and D (6, 1) are the positions of these 4 friends.
AB = `sqrt((3-6)^2+(4-7)^2)`
= `sqrt((-3)^2+(-3)^2)`
= `sqrt(9+9)`
= `sqrt18`
= `3sqrt2`
BC = `sqrt((6-9)^2+(7-4)^2) `
= `sqrt((-3)^2+(3)^2)`
= `sqrt(9+9)`
= `sqrt18`
= `3sqrt2`
CD = `sqrt((9-6)^2+(4-1)^2)`
= `sqrt((3)^2+(3)^2)`
= `sqrt(9+9)`
= `sqrt18`
= `3sqrt2`
AD = `sqrt((3-6)^2+(4-1)^2)`
= `sqrt((-3)^2 + (3)^2)`
= `sqrt(9+9)`
= `sqrt18`
= `3sqrt2`
Diagonal AC = `sqrt((3-9)^2+(4-4)^2)`
= `sqrt((-6)^2)`
= 6
Diagonal BD = `sqrt((6-6)^2+(7-1)^2)`
= `sqrt((6)^2)`
= 6
It can be observed that all sides of this quadrilateral ABCD are of the same length and also the diagonals are of the same length.
Therefore, ABCD is a square and hence, Champa was correct.
संबंधित प्रश्न
Find the distance between the following pairs of points:
(2, 3), (4, 1)
Find the distance between the following pairs of points:
(a, b), (−a, −b)
Find the co-ordinates of points of trisection of the line segment joining the point (6, –9) and the origin.
Find the distance between the following pair of point.
T(–3, 6), R(9, –10)
Determine whether the points are collinear.
L(–2, 3), M(1, –3), N(5, 4)
Find the distance between the following pairs of point in the coordinate plane :
(7 , -7) and (2 , 5)
Find the distance of the following point from the origin :
(8 , 15)
Find the distance between the following point :
(p+q,p-q) and (p-q, p-q)
Find the distance between the following point :
(Sin θ - cosec θ , cos θ - cot θ) and (cos θ - cosec θ , -sin θ - cot θ)
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.
Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles right angled triangle.
Prove that the points (5 , 3) , (1 , 2), (2 , -2) and (6 ,-1) are the vertices of a square.
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.
Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
Find distance between points O(0, 0) and B(– 5, 12)
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 P(–6, 8) from the origin is ______.
Name the type of triangle formed by the points A(–5, 6), B(–4, –2) and C(7, 5).
