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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
If A(4, 3), B(-1, y) and C(3, 4) are the vertices of a right triangle ABC, right-angled at A, then find the value of y.
If the opposite vertices of a square are (1, – 1) and (3, 4), find the coordinates of the remaining angular points.
Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
Prove that the points A(1, 7), B (4, 2), C(−1, −1) D (−4, 4) are the vertices of a square.
Using the distance formula, show that the given points are collinear:
(-1, -1), (2, 3) and (8, 11)
Find the distance between the following pairs of point in the coordinate plane :
(4 , 1) and (-4 , 5)
Find the distance between the following pairs of point in the coordinate plane :
(13 , 7) and (4 , -5)
Find the distance of the following point from the origin :
(8 , 15)
Find the distance of a point (7 , 5) from another point on the x - axis whose abscissa is -5.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
Prove that the points (7 , 10) , (-2 , 5) and (3 , -4) are vertices of an isosceles right angled triangle.
Prove that the points (a, b), (a + 3, b + 4), (a − 1, b + 7) and (a − 4, b + 3) are the vertices of a parallelogram.
Find the distance between the following pairs of points:
`(3/5,2) and (-(1)/(5),1(2)/(5))`
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
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.
If the point (x, y) is at equidistant from the point (a + b, b – a) and (a-b, a + b). Prove that ay = bx.
Find distance between points O(0, 0) and B(– 5, 12)
Show that P(– 2, 2), Q(2, 2) and R(2, 7) are vertices of a right angled triangle
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:
If a player P needs to be at equal distances from A and G, such that A, P and G are in straight line, then position of P will be given by ______.
