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In the figure, ABCD is a square, BC║GE, GE/EF = 2/3, and 9=CF. Find the area of ABCD

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.

Learn how to find the area of square ABCD using geometric conditions involving similar triangles and side length relationships. This problem covers key concepts in geometry, including the AA similarity theorem and area calculations. Suitable for high school students.

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