A circle is inscribed in a square. If the length of the diagonal of the square is 14√2 cm, what is the area (in square centimetres, using π = 22/7) of the inscribed circle?

Introduction / Context:
This question links a square and an inscribed circle. The circle touches all four sides of the square internally, so the diameter of the circle is equal to the side length of the square. The problem provides the diagonal of the square and asks for the area of the circle. To solve this, we use the relationship between the side and the diagonal of a square, and then apply the area formula for a circle with a specified value of π.

Given Data / Assumptions:
- A circle is inscribed in a square.- Diagonal of the square = 14√2 cm.- The circle touches all sides of the square internally.- π is to be taken as 22/7.- Area of a circle is given by Area = π * r^2.

Concept / Approach:
In a square of side s, the diagonal is s * √2. Here, the diagonal is given, so we can find the side s. For a circle inscribed in a square, the diameter of the circle equals the side length of the square. Once we know the side length, we know the diameter, and the radius is half of that. Then we apply the standard circle area formula using the given value of π. This requires correct handling of square roots and careful substitution.

Step-by-Step Solution:
Step 1: Let the side of the square be s cm.Step 2: The diagonal of a square is given by diagonal = s * √2.Step 3: We are given that diagonal = 14√2 cm.Step 4: Therefore, s * √2 = 14√2, which implies s = 14 cm.Step 5: The circle is inscribed in the square, so the diameter of the circle = side of the square = 14 cm.Step 6: The radius r of the circle = diameter / 2 = 14 / 2 = 7 cm.Step 7: Area of the circle = π * r^2.Step 8: Substitute r = 7 cm: Area = π * 7^2 = π * 49.Step 9: Using π = 22/7, Area = 49 * 22 / 7.Step 10: Simplify 49 / 7 = 7, so Area = 7 * 22 = 154 square centimetres.

Verification / Alternative check:
We can quickly check by noting that a circle of radius 7 cm usually has area 49π. With π approximated as 3.14, 49 * 3.14 ≈ 153.86, which is very close to 154. Since the problem explicitly states to use π = 22/7, 154 is exact. Additionally, confirming that side = 14 from the diagonal 14√2 is straightforward, ensuring there is no algebraic slip. Thus, the calculated area is consistent and correct.

Why Other Options Are Wrong:
- 308: This would be 2 * 154 and would correspond to doubling the correct area, perhaps from misinterpreting diameter as radius.- 462: This is three times 154 and has no direct geometric basis in this problem.- 616: This is four times 154, again indicating overcounting or misusing radius and diameter.- 231: This does not correspond to any correct combination of radius or diameter with π = 22/7 in this configuration.

Common Pitfalls:
One common error is to confuse the diagonal with the side of the square or to forget the √2 factor in the diagonal formula. Another is to mistake the diameter for the radius and plug 14 instead of 7 into the formula for r^2, which would give an area four times too large. Also, failing to use the specified value of π (22/7) can lead to answers that do not match the given options exactly. Careful reading and correct application of formulas help avoid these issues.

Final Answer:
The area of the inscribed circle is 154 square centimetres.