What is the area of the square?

Statement I: The length of the diagonal of the square is equal to the diameter of a circle.

Statement II: The area of the circle circumscribing the square is 154 sq.m.

Question

What is the area of the square?

Statement I: The length of the diagonal of the square is equal to the diameter of a circle.

Statement II: The area of the circle circumscribing the square is 154 sq.m.

Accepted Answer

**Solution:**

1. From Statement I:
   - The diagonal of the square equals the diameter of a circle.
   - However, without knowing the actual length of the diagonal or diameter, we cannot find the area of the square.
   - So, Statement I alone is insufficient.

2. From Statement II:
   - The area of the circle circumscribing the square is given as 154 sq.m.
   - The circle circumscribing the square means the circle passes through all four vertices of the square.
   - The radius of this circle is half the diagonal of the square.
   - But without the diagonal length or side length, we cannot directly find the square's area.
   - So, Statement II alone is insufficient.

3. Combining both statements:
   - From Statement II, area of circle = 154 sq.m.
   - Area of circle = $ \pi r^2 = 154 $
   - Using $ \pi = \frac{22}{7} $,
     $$ r^2 = \frac{154 \times 7}{22} = 49 \implies r = 7 \text{ m} $$
   - Diameter = $ 2r = 14 \text{ m} $
   - From Statement I, diagonal of square = diameter of circle = 14 m
   - Side of square $ s = \frac{\text{diagonal}}{\sqrt{2}} = \frac{14}{\sqrt{2}} = 7\sqrt{2} \text{ m} $
   - Area of square $ = s^2 = (7\sqrt{2})^2 = 49 \times 2 = 98 \text{ sq.m} $

**Answer:** Both statements together are necessary to find the area of the square.

Answer

Answer

Answer

Answer