The perimeter of a rectangle and a square are 160 m each. The area of the rectangle is less than that of the square by 100 sq. m. The length of the rectangle is

### Concept & Quadratic Equations in Geometry When given the sum and product of two variables (like length and breadth), it naturally forms a quadratic equation. The sum is derived from the perimeter, and the product from the area. $$ l + b = \frac{\text{Perimeter}}{2} $$ $$ l \times b = \text{Area} $$ ### Step-by-Step Solution * The perimeter of the square is 160 m. Its side is $160 / 4 = 40$ m. * The area of the square is $40^2 = 1600$ sq m. * The perimeter of the rectangle is also 160 m: $2(l + b) = 160 \Rightarrow l + b = 80$ m. * The area of the rectangle is 100 sq m less than the square's area: $1600 - 100 = 1500$ sq m. Thus, $l \times b = 1500$. * We need two numbers that add up to 80 and multiply to 1500. By inspecting the factors, these are clearly 50 and 30. * Since the length is typically considered the longer dimension, $l = 50$ m and $b = 30$ m. ### Exam Strategy & Shortcut Once you know $l + b = 80$ and $l \times b = 1500$, look at the options. You need an option that, when subtracted from 80, gives a number that multiplies with the option to yield 1500. $50 \times (80 - 50) = 50 \times 30 = 1500$. Option (c) is immediately correct. ### Common Pitfall A common mistake is misreading the phrase "less than that of the square by 100" and instead adding 100 to the square's area, which leads to incorrect factors and a confusing equation. ### Final Answer Therefore, the correct answer is **50 m**.