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The problem involves two triangles that are similar, and we need to find the two possible values of \( x \), the length of side \( BC \).

### **Assumption 1:**
Triangles \( ABE \) and \( ACD \) are similar. Since the triangles are similar, their corresponding sides are proportional.

Using the given lengths:
- \( AB = 10 \, \text{cm} \)
- \( BE = 3 \, \text{cm} \)
- \( AE = 15 \, \text{cm} \)
- \( AC = x \, \text{cm} \)
- \( CD = 3 \, \text{cm} \)

For similar triangles, the ratio of corresponding sides is equal, so:
\[
\frac{AB}{AC} = \frac{BE}{CD}
\]
Substitute the known values:
\[
\frac{10}{x} = \frac{3}{3}
\]
Simplifying:
\[
\frac{10}{x} = 1
\]
Thus:
\[
x = 10 \, \text{cm}
\]

### **Assumption 2:**
Now assume that triangles \( ABE \) and \( ABC \) are similar (smaller triangle within the larger triangle). Again, use the proportionality of corresponding sides.

We know:
- \( AB = 10 \, \text{cm} \)
- \( BE = 3 \, \text{cm} \)
- \( AE = 15 \, \text{cm} \)
- \( AC = x \, \text{cm} \)
- \( BC = 3 \, \text{cm} \)

The ratio of corresponding sides:
\[
\frac{AB}{AC} = \frac{BE}{BC}
\]
Substitute the values:
\[
\frac{10}{x} = \frac{3}{3}
\]
Simplifying:
\[
x = 2 \, \text{cm}
\]

Thus, the two possible values for \( x \) are 2 cm and 10 cm.

Would you like further details or clarifications on any of these steps?

Here are some related questions:
1. What are the properties of similar triangles?
2. How do we know that triangles \( ABE \) and \( ACD \) are similar?
3. Can you explain the difference between scaling factors for similar triangles?
4. What is the significance of corresponding sides in similar triangles?
5. How would the problem change if one of the angles were different?

**Tip**: Always check if angles are equal when assuming triangles are similar. Angle-Angle similarity can be a useful approach.

The two triangles in the diagram are similar. There are two possible values of x. Work out each of these values. State both assumptions you make in your working.

Find the two possible values of x in the problem involving similar triangles ABE and ACD. This solution covers proportionality, theorems of similar triangles, and step-by-step reasoning for Grade 9-10 students.

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