Question

What is the difference between the lengths of any two distinct sides of the rectangle? Statement (I): The perimeter of the rectangle is equal to that of a circle of radius \(\frac{10}{\pi}\) cm. Statement (II): The area of the rectangle is equal to that of a square of side \(4\) cm.

• A. Statement I alone is sufficient, but Statement II alone is not sufficient.
• B. Statement II alone is sufficient, but Statement I alone is not sufficient.
• C. Both statements together are sufficient, but neither statement alone is sufficient.
• D. Even both statements together are not sufficient.

Hint:

For rectangles, knowing only perimeter or only area is usually insufficient. Together they often determine the dimensions uniquely.

Answer 1

The Correct Option is C

Concept: Let the sides of the rectangle be \(l\) and \(b\). Then \[ \text{Perimeter}=2(l+b) \] and \[ \text{Area}=lb. \] To determine \(l-b\), we generally need both the sum and the product of the sides.

Step 1: Analyze Statement (I). Circumference of the circle: \[ 2\pi r = 2\pi\left(\frac{10}{\pi}\right) = 20. \] Hence rectangle perimeter is \[ 20. \] Therefore, \[ 2(l+b)=20 \] \[ l+b=10. \] Only the sum is known. Statement (I) alone is insufficient.

Step 2: Analyze Statement (II). Area of square: \[ 4^2=16. \] Thus, \[ lb=16. \] Only the product is known. Statement (II) alone is insufficient.

Step 3: Combine both statements. We have \[ l+b=10 \] and \[ lb=16. \] Therefore \[ x^2-10x+16=0. \] Factoring, \[ (x-8)(x-2)=0. \] Thus sides are \[ 8 \text{ cm and } 2 \text{ cm}. \] Difference: \[ 8-2=6. \] A unique value is obtained. Hence both statements together are sufficient.