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A rectangle has one side that is 6 cm shorter than the other...

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

A.

15 cm

B.

19 cm

C.

13 cm

D.

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