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प्रश्न
Show that the points (0, –1), (8, 3), (6, 7) and (–2, 3) are vertices of a rectangle.
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उत्तर
Let the points be P(0, –1), Q(8, 3), R(6, 7), S(–2, 3)
Distance between two points = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
∴ By distance formula,
d(P, Q) = `sqrt((8 - 0)^2 + [3 - (-1)]^2`
= `sqrt((8 - 0)^2 + (3 + 1)^2`
= `sqrt(8^2 + 4^2)`
= `sqrt(64 + 16)`
= `sqrt(80)` ...(i)
d(Q, R) = `sqrt((6 - 8)^2 + (7 - 3)^2`
= `sqrt((-2)^2 + (4)^2`
= `sqrt(4 + 16)`
= `sqrt(20)` ...(ii)
d(R, S) = `sqrt([(-2) - 6]^2 + (3 - 7)^2`
= `sqrt((-8)^2 + (-4)^2`
= `sqrt(64 + 16)`
=`sqrt(80)` ...(iii)
d(P, S) = `sqrt([(-2) - 0]^2 + [3 - (-1)^2]`
= `sqrt((-2)^2+ (3+ 1)^2`
= `sqrt((-2)^2 + 4^2`
= `sqrt(4 + 16)`
= `sqrt(20)` ...(iv)
In ▢PQRS,
∴ side PQ = side RS ...[From (i) and (iii)]
side QR = side PS ...[From (ii) and (iv)]
∴ ▢PQRS is a parallelogram ...[A quadrilateral is a parallelogram, if both the pairs of its opposite sides are congruent]
d(P, R) = `sqrt((6 - 0)^2 + [7 - (-1)]^2`
= `sqrt((6 - 0)^2 + (7 + 1)^2`
= `sqrt(6^2 + 8^2)`
= `sqrt(36 + 64)`
= `sqrt(100)`
= 10 ...(iv)
d(Q, S) = `sqrt([(-2) - 8]^2 + [3 - 3]^2`
= `sqrt((-10)^2 + (0)^2`
= `sqrt(100 + 0)`
= `sqrt(100)`
= 10 ...(vi)
In parallelogram PQRS,
PR = QS ...[From (v) and (vi)]
∴ ▢PQRS is a rectangle. ...[A parallelogram is a rectangle if its diagonals are equal]
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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.
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(−3, 5), (3, 1), (0, 3), (−1, −4)
If Q (0, 1) is equidistant from P (5, –3) and R (x, 6), find the values of x. Also, find the distances QR and PR.
Show that the points A (1, –2), B (3, 6), C (5, 10) and D (3, 2) are the vertices of a parallelogram.
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
Two opposite vertices of a square are (–1, 2) and (3, 2). Find the coordinates of other two vertices.
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 co-ordinates of points of trisection of the line segment joining the point (6, –9) and the origin.
Find all possible values of x for which the distance between the points A(x, –1) and B(5, 3) is 5 units.
If P(x, y) is equidistant from the points A(7, 1) and B(3, 5), find the relation between x and y.
Find the relation between a and b if the point P(a ,b) is equidistant from A (6,-1) and B (5 , 8).
Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius.
The centre of a circle passing through P(8, 5) is (x+l , x-4). Find the coordinates of the centre if the diameter of the circle is 20 units.
Calculate the distance between A (7, 3) and B on the x-axis, whose abscissa is 11.
Find distance between point Q(3, –7) and point R(3, 3)
Solution: Suppose Q(x1, y1) and point R(x2, y2)
x1 = 3, y1 = –7 and x2 = 3, y2 = 3
Using distance formula,
d(Q, R) = `sqrt(square)`
∴ d(Q, R) = `sqrt(square - 100)`
∴ d(Q, R) = `sqrt(square)`
∴ d(Q, R) = `square`
The distance of the point (α, β) from the origin 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 ______.
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 x axis equidistant from I and E is ______.
Find the value of a, if the distance between the points A(–3, –14) and B(a, –5) is 9 units.
