The area of a rectangle is 1200 square centimetres and its perimeter is 140 centimetres. What is the length of the diagonal of this rectangle?

Introduction / Context:
This problem combines several key properties of rectangles: the relationships between area, perimeter, side lengths, and the diagonal. It requires algebraic manipulation of two equations to find the side lengths, and then application of the Pythagoras theorem to find the diagonal. Such multi-step questions are good practice for translating word statements into equations and connecting different geometric formulas.

Given Data / Assumptions:

• Let the rectangle have length L and breadth B (both in cm).
• Area A = L * B = 1200 square centimetres.
• Perimeter P = 2(L + B) = 140 centimetres.
• We must find the length of the diagonal d of the rectangle.
• We will use the Pythagoras theorem: d^2 = L^2 + B^2.

Concept / Approach:
From the perimeter, we can express L + B. From the area, we know L * B. We thus have a system of two equations in L and B. Once we solve this system, we apply the Pythagoras theorem to find the diagonal d. Rectangles always have right angles, so the diagonal is the hypotenuse of a right triangle with legs L and B.

Step-by-Step Solution:
Step 1: Use the perimeter: 2(L + B) = 140 ⇒ L + B = 70. Step 2: Use the area: L * B = 1200. Step 3: Treat L and B as roots of a quadratic equation t^2 - (sum)t + product = 0. Step 4: So t^2 - 70t + 1200 = 0. Step 5: Factor or solve using the discriminant. Compute the discriminant: D = 70^2 - 4 * 1200 = 4900 - 4800 = 100. Step 6: Roots are: t = [70 ± √100] / 2 = [70 ± 10] / 2. Step 7: Thus L and B are 40 cm and 30 cm (order does not matter). Step 8: Now use the Pythagoras theorem for the diagonal: d^2 = 40^2 + 30^2 = 1600 + 900 = 2500. Step 9: Therefore d = √2500 = 50 cm.

Verification / Alternative check:
Verify by recomputing area and perimeter. With L = 40 cm and B = 30 cm, area A = 40 * 30 = 1200 square centimetres, which matches the given area. Perimeter P = 2(40 + 30) = 2 * 70 = 140 centimetres, also matching the problem statement. The diagonal of a 3-4-5 scaled triangle (here 30, 40, 50) is well known to be 50 cm, which further confirms our calculations.

Why Other Options Are Wrong:
If the diagonal were 30 cm or 40 cm, then at least one side would have to be shorter than is possible given the area and perimeter constraints. A diagonal of 60 cm would come from larger sides, producing a bigger area than 1200 square centimetres. A diagonal of 70 cm is even more extreme and would not be consistent with the given perimeter. Only 50 cm matches the side lengths determined from the simultaneous equations.

Common Pitfalls:
Mistakes often occur when setting up the quadratic equation, especially in writing the correct product term or computing the discriminant. Another common error is to guess side lengths without checking both area and perimeter simultaneously. Some students also misuse the Pythagoras theorem by adding perimeter values or mixing up which quantities are squared. Working systematically from equations and checking each step reduces these errors significantly.

Final Answer:
The length of the diagonal of the rectangle is 50 cm.