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The diagram shows two triangles that are similar, and we are tasked with finding two possible values of \( x \), representing the side \( BC \). The problem also specifies that we need to make two assumptions about which triangles are similar in each case.

### Assumption 1
Let's assume the two triangles \(\triangle ABC\) and \(\triangle ADE\) are similar.

In this assumption, the ratio of the corresponding sides of the triangles must be equal. The corresponding sides are:

\[
\frac{AB}{AE} = \frac{BC}{ED} = \frac{AC}{AD}
\]

From the diagram:
- \( AB = 10 \) cm,
- \( AE = 15 \) cm,
- \( BC = x \) cm,
- \( ED = 3 \) cm,
- \( AC = 10 + x \) cm,
- \( AD = 18 \) cm.

So, applying the ratios from the similar triangles:
\[
\frac{10}{15} = \frac{x}{3}
\]
Solving for \( x \):
\[
\frac{10}{15} = \frac{2}{3}, \quad \frac{x}{3} = \frac{2}{3}
\]
Thus:
\[
x = 2 \, \text{cm}.
\]

### Assumption 2
Now, let’s assume the triangles \(\triangle ABC\) and \(\triangle ABE\) are similar.

In this case, the ratio of corresponding sides becomes:
\[
\frac{AB}{AE} = \frac{AC}{AD}
\]

Using the known values:
- \( AB = 10 \) cm,
- \( AE = 15 \) cm,
- \( AC = 10 + x \) cm,
- \( AD = 18 \) cm.

So the ratio equation is:
\[
\frac{10}{15} = \frac{10 + x}{18}
\]
Simplifying:
\[
\frac{2}{3} = \frac{10 + x}{18}
\]
Cross-multiplying to solve for \( x \):
\[
2 \times 18 = 3 \times (10 + x)
\]
\[
36 = 30 + 3x
\]
\[
36 - 30 = 3x
\]
\[
6 = 3x
\]
\[
x = 2 \, \text{cm}.
\]

### Answers:
- Under Assumption 1, \( x = 2 \) cm.
- Under Assumption 2, \( x = 2 \) cm.

If you'd like further explanation or have questions about this solution, feel free to ask!

Here are 5 questions you can explore:
1. Why do we assume that the triangles are similar in both cases?
2. What is the importance of corresponding sides in similar triangles?
3. Can the concept of similar triangles be applied to other geometrical figures?
4. How would the problem change if one of the sides was unknown?
5. How would you calculate the area of the triangles once \( x \) is known?

**Tip:** Always check the proportionality of corresponding sides when using similar triangles to ensure the setup is correct.

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.

Learn how to find two possible values of x in a triangle diagram where similar triangles are used. This problem involves setting up ratios of corresponding sides and solving algebraic equations. Suitable for Grades 9-10 students.

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