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Question
Show that the ▢PQRS formed by P(2, 1), Q(–1, 3), R(–5, –3) and S(–2, –5) is a rectangle.
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Solution
Given: P(2, 1), Q(–1, 3), R(–5, –3) and S(–2, –5)
Distance Formula = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
PQ = `sqrt((-1 - 2)^2 + (3 - 1)^2)`
= `sqrt((3-)^2 + (2)^2)`
= `sqrt(9 + 4)`
= `sqrt 13` ...(i)
QR = `sqrt([-5 - (-1)]^2 + (-3 - 3)^2)`
= `sqrt((-4)^2 + (-6)^2)`
= `sqrt(16 + 36)`
= `sqrt 52`
= `sqrt(2 xx 2 xx 13)`
= 2`sqrt13` ...(ii)
RS = `sqrt([-2 - (-5)]^2 + [-5 - (-3)]^2)`
= `sqrt((-2 + 5)^2 + (-5 + 3)^2)`
= `sqrt(3^2 + (-2)^2)`
= `sqrt(9 + 4)`
= `sqrt 13` ...(iii)
PS = `sqrt((-2 - 2)^2 + (-5 - 1)^2)`
= `sqrt((-4)^2 + (-6)^2)`
= `sqrt(16 + 36)`
= `sqrt52`
= `sqrt(2 xx 2 xx 13)`
= 2`sqrt 13` ...(iv)
In ▢PQRS,
PQ = RS ...[From (i) and (iii)]
QR = PS ...[From (ii) and (iv)]
∴ ▢PQRS is a parallelogram ...(A quadrilateral is a parallelogram if its opposite sides are equal)
By distance formula,
PR = `sqrt((-5 - 2)^2 + (-3 - 1)^2)`
= `sqrt((-7)^2 + (-4)^2)`
= `sqrt(49 +16)`
= `sqrt 65` ...(v)
QS = `sqrt([-2 - (-1)]^2 + (-5 - 3)^2)`
= `sqrt((-7)^2 + (-4)^2)`
= `sqrt(1 + 64)`
= `sqrt 65` ...(vi)
In parallelogram PQRS,
PQ = QS ...[From (v) and (vi)]
∴ ▢PQRS is a rectangle ...(A parallelogram is a rectangle, if its diagonals are equal.)
P(2, 1), Q(–1, 3), R(–5, –3) and S(–2, –5) are the vertices of a rectangle.
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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.
