In the figure shown above (a 10 cm × 10 cm square with successively smaller squares each formed by joining the midpoints of the sides of the next larger square, the innermost being shaded grey), each inside square is formed by joining the midpoints of the sides of the next larger square. The area of the smallest square (shaded) as shown, in cm², is:
Options:
(A) 12.50
(B) 6.25
(C) 3.125
(D) 1.5625
Official Answer Key:
C
Step-by-Step Solution & Explanation
Joining the midpoints of a square's sides produces a new (rotated) square whose area is exactly HALF the area of the parent square (side of new square = (side/√2), so area = side²/2). Starting from the outer 10 cm × 10 cm square (area = 100 cm²), each successive nested square halves the area: 100 → 50 → 25 → 12.5 → 6.25 → 3.125 cm². The figure shows the outer square followed by four more nested squares down to the shaded one (five halving steps in total), giving 100 × (1/2)^5 = 3.125 cm², matching option (C).
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