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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.
Find the distance between two points
(i) P(–6, 7) and Q(–1, –5)
(ii) R(a + b, a – b) and S(a – b, –a – b)
(iii) `A(at_1^2,2at_1)" and " B(at_2^2,2at_2)`
If Q (0, 1) is equidistant from P (5, − 3) and R (x, 6), find the values of x. Also find the distance QR and PR.
If the points (2, 1) and (1, -2) are equidistant from the point (x, y), show that x + 3y = 0.
A(–8, 0), B(0, 16) and C(0, 0) are the vertices of a triangle ABC. Point P lies on AB and Q lies on AC such that AP : PB = 3 : 5 and AQ : QC = 3 : 5. Show that : PQ = `3/8` BC.
Find the distance of the following points from the origin:
(ii) B(-5,5)
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 :
(5 , 12)
Find the distance of the following point from the origin :
(6 , 8)
Find the distance of a point (7 , 5) from another point on the x - axis whose abscissa is -5.
Find the distance of a point (12 , 5) from another point on the line x = 0 whose ordinate is 9.
Find the value of m if the distance between the points (m , -4) and (3 , 2) is 3`sqrt 5` units.
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 (6 , -1) , (5 , 8) and (1 , 3) are the vertices of an isosceles triangle.
Prove that the points A (1, -3), B (-3, 0) and C (4, 1) are the vertices of an isosceles right-angled triangle. Find the area of the triangle.
The distance between point P(2, 2) and Q(5, x) is 5 cm, then the value of x ______
The point which lies on the perpendicular bisector of the line segment joining the points A(–2, –5) and B(2, 5) 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:
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 ______.
Name the type of triangle formed by the points A(–5, 6), B(–4, –2) and C(7, 5).
|
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.
