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In this problem, we are given that quadrilateral **ABCD** is similar to quadrilateral **FECG**. To find the value of \( y \), we will leverage the properties of similar figures. Here are the steps:

### Step 1: Properties of Similarity
Since **ABCD** is similar to **FECG**, the corresponding sides of these quadrilaterals will be proportional. Specifically, we can write the proportion between the corresponding sides as:
\[
\frac{AB}{FE} = \frac{BC}{EC} = \frac{CD}{CG} = \frac{AD}{FG}
\]

### Step 2: Identifying Corresponding Sides
From the diagram:
- \( AB = 5 \), \( FE = 6 \)
- \( BC = 5 \), \( EC = 5 \)
- \( CD = ? \), \( CG = 4 \)
- \( AD = y \), \( FG = 4 \)

Using these pairs, we can set up a proportion to solve for \( y \).

### Step 3: Set Up the Proportion for \( y \)
We know that:
\[
\frac{AD}{FG} = \frac{AB}{FE}
\]
Substituting the values:
\[
\frac{y}{4} = \frac{5}{6}
\]

### Step 4: Solve for \( y \)
Now, solve for \( y \) by cross-multiplying:
\[
y = \frac{5 \times 4}{6} = \frac{20}{6} = \frac{10}{3} \approx 3.33
\]

### Final Answer:
Thus, the value of \( y \) is approximately \( 3.33 \).

Would you like further details on this solution, or do you have any additional questions?

### 5 Related Questions:
1. How do we prove that two figures are similar?
2. Can the angles in similar polygons help in solving problems like this one?
3. What are other ways to use the properties of proportionality in geometry?
4. How do we verify if two given shapes are congruent versus similar?
5. How can we apply the Pythagorean theorem to solve problems involving similar triangles?

### Tip:
When dealing with similar figures, always match corresponding sides carefully and use the proportionality rules to solve for unknown lengths.

Given quadrilateral ABCD is similar to quadrilateral FECG, find the value of y using properties of similar figures and proportional sides.

Solve a problem involving similar quadrilaterals ABCD and FECG by applying the properties of similar figures and proportional sides. This problem uses key geometric concepts such as proportionality and similarity theorems. Suitable for high school geometry students.

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