Question

length of a rectangle is 5 less than twice its breath. if the perimeter is 110 than find area of rectangle with cross multiplication.

Answer 1

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• Perimeter (p) = 110 cm
• Let length of rectangle be 'x' and breadth be 'y'
• x = 2(y) - 5
• x - 2y + 5 = 0 --(i)

        Also:

• Perimeter (p) = 2 (length + breadth)
• 110 = 2 (x + y)
• x + y = $\bold{\frac{110}{2} }$
• x + y = 55
• x + y - 55 = 0 --(ii)

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• $\bold{\dfrac{x}{b_1 c_2 - b_2 c_1} = \dfrac{y}{c_1 a_2 - c_2 a_1} = \dfrac{1}{a_1 b_2 - a_2 b_1}}$

• x =  $\bold{\dfrac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1}}$

• y = $\bold{\dfrac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1}}$

$\bold{\blue{\underline{\red{Q}\pink{uest}\green{ion}\purple{:-}}}}$

• Find values of 'x' and 'y' by cross multiplication method and then find the area of rectangle .

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• a₁ = 1
• a₂ = 1
• b₁ = -2
• b₂ = 1
• c₁ = 5
• c₂ = -55

• x =  $\bold{\dfrac{b_1 c_2 - b_2 c_1}{a_1 b_2 - a_2 b_1}}$  

• x = $\bold{\dfrac{-2\:\times\:-55 - 1\:\times\:5}{1\:\times\:-2 - 1\:\times\:1}}$

• x = $\bold{\dfrac{110\: - \: 5}{-2\:-\:1}}$

• x  = $\bold{\dfrac{105}{-3}}$

• x = 35 cm

• $\because$ Length is always measured positive as its a scalar quantity .

• y = $\bold{\dfrac{c_1 a_2 - c_2 a_1}{a_1 b_2 - a_2 b_1}}$

• y = $\bold{\dfrac{5\:\times\: 1\: - \:(-55)\:\times\: 1 }{1\:\times\:-2\:-\:1\:\times\:1}}$

• y = $\bold{\dfrac{5\: +\:55}{-2\:-\:1}}$  

• y = $\bold{\frac{60}{-3} }$

• y = 20 cm

• $\because$ Breadth is always measured positive as its a scalar quantity .

   $\bold{\blue{\underline{\red{To}\pink{\:find}\green{\:area}\purple{:}}}}$

• Area = length × breadth
• Area = x(y)
• Area = 35 × 20
• Area = 700 cm²

Answer 2

Let, the breadth be x

Perimeter of rectangle = 110

→ 2 ( l + b ) = 110

→ 2 ( 2x - 5 + x ) = 110

→ 2 ( 3x - 5 ) = 110

→ 6x - 10 = 110

→ 6x = 110 + 10

→ 6x = 120

→ x = 120/6

→ x = 20

Area of rectangle = l × b

⇒ Area of rectangle = ( 2 × 20 - 5 ) × 20

⇒ Area of rectangle = 40 - 5 × 20

⇒ Area of rectangle = 35 × 20

⇒ Area of rectangle = 35 × 20

⇒ Area of rectangle = 700