The diagonal of a rectangle is equal to square root of 41 centimetres and its area is 20 square centimetres. What is the perimeter of the rectangle, in centimetres?

Introduction / Context:
This problem involves a rectangle for which the diagonal and area are given. We are required to find the perimeter. This connects three basic relationships: the Pythagoras relation between sides and diagonal, the formula for area, and the formula for perimeter. Problems of this type test the ability to translate given data into equations and solve them systematically, often using algebraic identities.

Given Data / Assumptions:

• Let the sides of the rectangle be a centimetres and b centimetres.
• Diagonal = square root of 41 centimetres, so diagonal squared = 41.
• Area of rectangle = 20 square centimetres, so a * b = 20.
• Need perimeter P = 2 * (a + b).

Concept / Approach:
For a rectangle:

• By Pythagoras theorem: a^2 + b^2 = diagonal^2.
• Given diagonal^2 = 41, so a^2 + b^2 = 41.
• Given area: a * b = 20.

We recall identity: (a + b)^2 = a^2 + b^2 + 2ab. Using this, we can directly find a + b once we know a^2 + b^2 and ab. Then we compute the perimeter as 2 * (a + b).

Step-by-Step Solution:
Step 1: Use diagonal information. a^2 + b^2 = 41. Step 2: Use area information. ab = 20. Step 3: Use identity for (a + b)^2. (a + b)^2 = a^2 + b^2 + 2ab. (a + b)^2 = 41 + 2 * 20 = 41 + 40 = 81. Step 4: Find a + b. a + b = square root of 81 = 9. Step 5: Compute perimeter. Perimeter P = 2 * (a + b) = 2 * 9 = 18 centimetres. So, the perimeter of the rectangle is 18 centimetres.

Verification / Alternative check:
For completeness, we can find actual values of a and b by solving quadratic equations, but that is not necessary. For instance, suppose a and b are roots of the quadratic x^2 - (a + b)x + ab = 0. That is x^2 - 9x + 20 = 0. The roots are x = 4 and x = 5. So sides are 4 and 5. Check: area = 4 * 5 = 20, diagonal^2 = 4^2 + 5^2 = 16 + 25 = 41, diagonal = square root of 41. Perimeter = 2 * (4 + 5) = 18, which matches our answer.

Why Other Options Are Wrong:
Option A (9 cm) is just a + b, not twice that value. Option C (20 cm) is the area, not the perimeter. Option D (41 cm) is diagonal squared, and Option E (16 cm) might come from confusing a side with the perimeter. None of these satisfy all given conditions together.

Common Pitfalls:
A common mistake is to compute a and b separately without using the identity, which increases the chance of algebra errors. Others may confuse diagonal with its square, leading to wrong equations. Remember the useful shortcut that when both a^2 + b^2 and ab are known, the sum a + b is found easily using (a + b)^2 = a^2 + b^2 + 2ab.

Final Answer:
The perimeter of the rectangle is 18 centimetres.