https://file.aiquickdraw.com/mathbot/file/1724927301673l166tlvf.jpg

To determine the values of \( x \) and \( y \) using similarity, let's analyze the triangles shown in the image. Based on the image, here’s a breakdown of what I observe:

1. There is a larger triangle, and inside it, there is a smaller triangle that shares one of its sides with the larger triangle. These triangles seem to be similar.

2. The given side lengths are:
   - \( DE = 7 \text{ cm} \)
   - \( BC = 10 \text{ cm} \)
   - \( AB = 12 \text{ cm} \)
   - \( AC = 9 \text{ cm} \)
   - \( BE = 3 \text{ cm} \)
   - \( BD = 4 \text{ cm} \)
   - \( CD = 5 \text{ cm} \)

Let's determine \( x \) and \( y \), where \( x = AD \) and \( y = EC \), assuming \( AD = x \) and \( EC = y \).

### Step 1: Identify Similar Triangles
The two triangles \( \triangle ABC \) and \( \triangle ADE \) are similar by the Angle-Angle (AA) similarity criterion because they share an angle at \( A \), and the corresponding angles at \( D \) and \( B \) are equal.

### Step 2: Set up the Proportion
Since the triangles are similar, the corresponding sides are proportional:
\[
\frac{AD}{AB} = \frac{DE}{BC} = \frac{AE}{AC}
\]

Using the sides:
\[
\frac{x}{12} = \frac{7}{10} = \frac{y}{9}
\]

### Step 3: Solve for \( x \)
From the proportion \( \frac{x}{12} = \frac{7}{10} \):
\[
x = \frac{7}{10} \times 12 = 8.4 \text{ cm}
\]

### Step 4: Solve for \( y \)
From the proportion \( \frac{y}{9} = \frac{7}{10} \):
\[
y = \frac{7}{10} \times 9 = 6.3 \text{ cm}
\]

### Final Answer:
- \( x = 8.4 \text{ cm} \)
- \( y = 6.3 \text{ cm} \)

Would you like more details or have any further questions?

Here are 5 related questions:
1. How do you prove that two triangles are similar?
2. What are the properties of similar triangles?
3. How can you use similarity to solve real-world problems?
4. What is the relationship between the areas of two similar triangles?
5. How does the concept of proportionality apply to other geometric shapes?

**Tip:** Remember, the ratio of corresponding sides in similar triangles is equal, which can simplify many geometric problems.

Learn how to determine the values of x and y using similarity of triangles. Detailed step-by-step solution involving the Angle-Angle (AA) similarity criterion. Suitable for high school students and those preparing for geometry exams.

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