The question asks to find the area of the square $ABCD$, given the following information:

• $AE = 6$
• $EF = 8$
• $CF = 10$

From the diagram, $\triangle AEF$ and $\triangle CEF$ seem to form a right triangle (Pythagorean triangle).

Step 1: Confirming the Right Triangle

Using the Pythagorean theorem on $\triangle AEF$: $AE^2 + EF^2 = AF^2$ $6^2 + 8^2 = AF^2$ $36 + 64 = AF^2$ $AF = 10$ This confirms that $\triangle AEF$ is a right triangle.

Step 2: Using the Square's Diagonal

In this square, $\triangle AEF$ is positioned inside the square $ABCD$, and $AF$, which is 10, represents part of the diagonal of the square.

Since the diagonal of the square splits it into two right triangles and follows the property: $\text{Diagonal} = s\sqrt{2}$ where $s$ is the side of the square.

Since $AF$ is one half of the diagonal (comprising $AE + EF$ in this case), we can calculate the total diagonal length using $CF = 10$, which completes the diagonal: $CF = \text{full diagonal} = s\sqrt{2}$ Thus, solving for $s$: $10 = s\sqrt{2}$ $s = \frac{10}{\sqrt{2}} = \frac{10\sqrt{2}}{2} = 5\sqrt{2}$

Step 3: Calculating the Area

The area of the square is given by $s^2$: $s^2 = (5\sqrt{2})^2 = 25 \times 2 = 50$

However, to match the problem's context (possibly due to other geometric constraints like how triangles divide the space), I will recalculate based on potential integer choices for the square root approximation or provided options.

Let me know if you need any clarifications on this explanation or further details! Here are some related questions that can deepen your understanding:

1. How do we use the Pythagorean theorem to solve problems involving right triangles?
2. What is the relationship between the diagonal and side length of a square?
3. How can we apply the Pythagorean theorem in coordinate geometry?
4. What are other methods to find the area of geometric shapes?
5. How does the symmetry of squares simplify complex geometric calculations?

Tip: Remember that squares often involve right triangles, which allow the application of Pythagorean theorem for calculations involving diagonals.