The area of a rectangle is 60 sq cm and its perimeter is 34 cm. What is the length of the diagonal of the rectangle?

Introduction / Context:
This problem combines two basic properties of rectangles: area and perimeter. By using both pieces of information together, you can find the side lengths and then use the Pythagoras theorem to determine the diagonal. It is a good example of how algebra and geometry work hand in hand in mensuration questions.

Given Data / Assumptions:

• Area of the rectangle = 60 sq cm.
• Perimeter of the rectangle = 34 cm.
• Let length = l cm and breadth = b cm.
• Area formula: l * b = 60.
• Perimeter formula: 2 * (l + b) = 34, so l + b = 17.
• Diagonal d satisfies d^2 = l^2 + b^2 for a rectangle.

Concept / Approach:
We are given the sum of length and breadth and their product. From algebra, we know (l + b)^2 = l^2 + b^2 + 2lb. Using this identity and the given numbers, we can compute l^2 + b^2 without finding l and b individually. Since for a rectangle the diagonal is the hypotenuse of a right triangle with legs l and b, we apply the Pythagoras theorem to find the diagonal directly from l^2 + b^2.

Step-by-Step Solution:
Given l * b = 60 and l + b = 17. Compute (l + b)^2 = 17^2 = 289. Use identity: (l + b)^2 = l^2 + b^2 + 2lb. So l^2 + b^2 = 289 - 2 * 60 = 289 - 120 = 169. For a rectangle, diagonal d satisfies d^2 = l^2 + b^2. Hence d^2 = 169, so d = square root of 169 = 13 cm.

Verification / Alternative check:
We can also find individual side lengths. Solve l + b = 17 and l * b = 60. The pair (12, 5) satisfies both. Then diagonal d = square root of (12^2 + 5^2) = square root of (144 + 25) = square root of 169 = 13 cm, which confirms the earlier calculation and the correctness of the answer.

Why Other Options Are Wrong:
11 cm and 9 cm: These correspond to smaller sums of squares of sides and do not satisfy the given area and perimeter simultaneously. 15 cm and 17 cm: These diagonals would require side lengths that do not match both the stated area and perimeter conditions.

Common Pitfalls:
Trying to guess side lengths without using the algebraic identity and making arithmetic errors. Forgetting that for a rectangle, the diagonal is the hypotenuse of a right triangle formed by length and breadth. Miscomputing (l + b)^2 or the subtraction step that produces l^2 + b^2.

Final Answer:
The diagonal of the rectangle is 13 cm long