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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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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:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
