The shorter side of a rectangle is 21 cm less than the longer side. The numerical value of its area is equal to 7 times the numerical value of its perimeter. What is the length (in cm) of its longer side?

Correct option is D

Given:
The shorter side of a rectangle is 21 cm less than the longer side. The numerical value of its area is equal to 7 times the numerical value of its perimeter.
Formula Used:
Area of rectangle = Length × Breadth
Perimeter of rectangle = 2 × (Length + Breadth)
Solution:
Let the longer side of the rectangle be x cm.
Then, the shorter side = x − 21 cm
Area of rectangle = x × (x − 21)
Perimeter of rectangle = 2(x + x − 21) = 2(2x − 21)
Area = 7 × Perimeter
​$$x(x - 21) = 7 \times 2(2x - 21) \\x^2 - 21x = 14(2x - 21) \\x^2 - 21x = 28x - 294 \\x^2 - 49x + 294 = 0 \\x = \frac{49 \pm \sqrt{49^2 - 4 \cdot 1 \cdot 294}}{2} \\= \frac{49 \pm \sqrt{2401 - 1176}}{2} \\= \frac{49 \pm \sqrt{1225}}{2} \\= \frac{49 \pm 35}{2} \\x = \frac{49 + 35}{2} = \frac{84}{2} = 42 \quad \text{or} \quad x = \frac{49 - 35}{2} = \frac{14}{2} = 7$$​​
Check values:
If x = 7 => shorter side = -14 (not valid)
So, x = 42 cm is valid
Length of longer side is 42 cm