To find the area of square $ABCD$, we need to determine the side length of the square. Let's analyze the figure and use the given information:

Step 1: Analyze the given information

• $ABCD$ is a square, so all sides are equal.
• $BC \parallel GE$, which implies that triangles $BCG$ and $EFG$ are similar by AA similarity (corresponding angles are equal).
• The ratio $\frac{GE}{EF} = \frac{2}{3}$ is provided.
• $CF = 9$.

Step 2: Set up the relationship based on the similarity of triangles

Since $\frac{GE}{EF} = \frac{2}{3}$, let's denote:

• $EF = x$
• $GE = \frac{2}{3}x$

Because $CF = 9$, and $CF = CE + EF$, we have: $CF = CE + EF = x + \frac{2}{3}x$ $9 = x + \frac{2}{3}x = \frac{5}{3}x$

Step 3: Solve for $x$

$\frac{5}{3}x = 9$ $x = 9 \times \frac{3}{5} = \frac{27}{5} = 5.4$

So, $EF = x = 5.4$, and $GE = \frac{2}{3}x = 3.6$.

Step 4: Determine the side length of the square $AB$

Since $ABCD$ is a square and $BC$ is one side, we need to find $BC$. Since $BC = GE + EF$, and we've already determined $GE$ and $EF$, we have: $BC = GE + EF = 3.6 + 5.4 = 9$

Therefore, the side length of square $ABCD$ is $BC = 9$.

Step 5: Calculate the area of square $ABCD$

The area of square $ABCD$ is given by: $\text{Area} = \text{side}^2 = 9^2 = 81$

Final Answer:

The area of square $ABCD$ is 81 square units.

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Do you want more details or have any questions? Here are some related questions:

1. What is the length of $DE$ in the figure?
2. If $GE$ is extended to meet $AB$ at a point $H$, what is the length of $GH$?
3. What would be the perimeter of square $ABCD$?
4. How do the properties of similar triangles help solve this problem?
5. If $CF$ was doubled, what would be the new area of square $ABCD$?
6. How does the ratio $\frac{GE}{EF} = \frac{2}{3}$ affect the solution?
7. Can you find the area of triangle $BGE$?
8. What is the length of diagonal $AC$ in square $ABCD$?

Tip: When dealing with geometric figures, identifying similar triangles can simplify the problem significantly.