In a rectangle ABCD the lengths of sides AB, BC, CD and DA are ( 5x + 2y + 2 )cm, (x + y + 4) cm, (2x + 5y - 7) cm, and (3x + 2y - 11) cm respectively Which of the following statements is / are true ?

One of the sides of the rectangle is 15 cm long

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Each diagonal of the rectangle is 39 cm long

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Perimeter of the rectangle is 102 cm

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Area of the rectangle is 560

c m^{2}

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Solution

The correct options are
A One of the sides of the rectangle is 15 cm long
B Each diagonal of the rectangle is 39 cm long
C Perimeter of the rectangle is 102 cm
AD=BC
AB=DC
$3 x + 2 y - 11 = x + y + 4$
$5 x + 2 y + 2 = 2 x + 5 y - 7$
$2 x + y = 15$ ..........................eq(1) multiplied by 3
$3 x - 3 y = - 9$ ..............................eq(2)
on adding both equationswe get
$6 x + 3 y = 45$
$3 x - 3 y = - 9$
_____________________
$9 x = 36$
x=4
puting x=4 in eq(1)
$2 \times 4 + y = 15$
$y = 7$
AB=5x+2y+2
$= 5 \times 4 + 2 \times 7 + 2 = 20 + 14 + 2 = 36$
BC=x+y+4
$= 4 + 7 + 4 = 15$
So, perimetre =2(36+15)
$= 102$
Area= $= 36 \times 15 = 540$
Diagonal= $= \sqrt{15^{2} + 36^{2}} = \sqrt{1521} = 39$
so a b c points are correct.

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