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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.
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.
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संबंधित प्रश्न
Find the coordinates of the circumcentre of the triangle whose vertices are (8, 6), (8, – 2) and (2, – 2). Also, find its circum radius
Find the distance between the following pairs of points:
(2, 3), (4, 1)
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(4, 5), (7, 6), (4, 3), (1, 2)
Show that the points A (1, −2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
Using the distance formula, show that the given points are collinear:
(6, 9), (0, 1) and (-6, -7)
Determine whether the point is collinear.
R(0, 3), D(2, 1), S(3, –1)
Find x if distance between points L(x, 7) and M(1, 15) is 10.
Find the distance of the following point from the origin :
(13 , 0)
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 relation between a and b if the point P(a ,b) is equidistant from A (6,-1) and B (5 , 8).
Prove that the following set of point is collinear :
(5 , 1),(3 , 2),(1 , 3)
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 origin and the point:
(-5, -12)
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
Find a point on the y-axis which is equidistant from the points (5, 2) and (-4, 3).
Prove that the points P (0, -4), Q (6, 2), R (3, 5) and S (-3, -1) are the vertices of a rectangle PQRS.
Calculate the distance between the points P (2, 2) and Q (5, 4) correct to three significant figures.
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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?
