Question

A square is drawn by joining the midpoints of the sides of a given square. A third square is drawn inside the second square in the same way and this process is continued indefinitely. If a side of the first square is 8 cm, the sum of the areas of all the squares thus formed (in sq.cm) is:

• A. 128
• B. 120
• C. 96
• D. None of these

Hint:

For geometric shrinkage patterns, check if each new figure’s area is a fixed fraction of the previous — then apply infinite GP sum formula.

Answer 1

The Correct Option is A

Area of first square = $8^2 = 64$ sq.cm.
When a square is drawn by joining midpoints of sides, the new square’s side = $\frac{\text{original side}}{\sqrt{2}}$.
Thus, area of new square = $\left( \frac{\text{side}}{\sqrt{2}} \right)^2 = \frac{\text{original area}}{2}$.
So areas form GP: $64 + 32 + 16 + $ to infinity.
Sum = $\frac{\text{first term}}{1 - r}$ with $r = \frac{1}{2}$:
$S = \frac{64}{1 - \frac{1}{2}} = \frac{64}{\frac{1}{2}} = 128$ sq.cm.