A rectangle has one side that is 6 cm shorter than the other. The area of the rectangle will increase by 68 cm $^2$ if we add 2 cm to each side of the rectangle. Find the length of the shorter side

15 cm

19 cm

13 cm

21 cm

Correct answer is C

Let the length of the longer side = $x$ cm

∴ The length of the shorter side = ( $x$ - 6) cm

If we increase each side's length by 2 cm, it becomes

( $x$ + 2) cm and ( $x$ - 4) cm respectively

Area of a rectangle = L x B

$A_1 = x(x - 6) = x^2 - 6x$

$A_2 = (x + 2)(x - 4) = x^2 - 4x + 2x - 8 = x^2 - 2x - 8$

$A_1 + 68 = A_2$ (Given)

⇒ $x^2 - 6x + 68 = x^2 - 2x - 8$

⇒ $x^2 - x^2 - 6x + 2x$ = -8 - 68

⇒ -4 $x$ = -76

⇒ $x$ = $\frac{-76}{-4}$ = 19cm

∴ The length of the shorter side = $x$ - 6 = 19 - 6 = 13 cm

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