Question

The adjoining figure shows a set of concentric squares. If the diagonal of the innermost square is 2 units, and if the distance between the corresponding corners of any two successive squares is 1 unit, find the difference between the areas of the eighth and the seventh squares, counting from the innermost square.

• A. \( 10\sqrt{2} \) sq. units
• B. \( 35\sqrt{2} \) sq. units
• C. 30 sq. units
• D. None of these

Hint:

Corresponding corners of successive squares lie on the same diagonal line, so each step outward adds 1 unit at both ends of a diagonal. Use area = (diagonal)\(^2\)/2.

Answer 1

The Correct Option is C

Step 1: Find the diagonals:
The squares are concentric and nested with the same orientation, so corresponding corners lie along a common diagonal through the centre. A step of 1 unit at each end increases the diagonal by 2 units. The diagonals are 2, 4, 6, ..., so the \( n \)th square has diagonal \( 2n \).

Step 2: Use the area formula:
Area of a square = \( \dfrac{d^2}{2} = \dfrac{(2n)^2}{2} = 2n^2 \).

Step 3: Compute the difference:
8th square: \( 2 \times 64 = 128 \). 7th square: \( 2 \times 49 = 98 \).
Difference \( = 128 - 98 = 30 \) sq. units.

Final Answer:
The correct option is (3): 30 sq. units