What is the area of the smaller square?
Consider the following for the next three (03) items that follow:<br>In the figure given below, a circle is inscribed in a square PQRS. A rectangle at the corner P that measures 4\text{ cm} \times 2\text{ cm} and a square at the corner R are drawn.
- A. 50(3-\sqrt{2}) square cm
- B. 25(3-2\sqrt{2}) square cm
- C. 25(3+2\sqrt{2}) square cm
- D. 50(3-2\sqrt{2}) square cm ✓
Correct Answer: D. 50(3-2\sqrt{2}) square cm
Explanation
Let the small square have side s. The diagonal distance from the corner to the circle's center is R\sqrt{2}. Thus, s\sqrt{2} + R = R\sqrt{2} \implies s = R(1 - 1/\sqrt{2}). For R=10, s = 10 - 5\sqrt{2}. The area s^2 = (10-5\sqrt{2})^2 = 50(3-2\sqrt{2}).
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