Let's examine the diagram and solve the problem.

3.5.1 Scale Factor

Since $\triangle ABC$ is similar to $\triangle DEF$ (denoted by $\triangle ABC ||| \triangle DEF$), we can determine the scale factor by comparing corresponding sides.

Given:

• $BC = 6$ and $EF = y$,
• $AC = 5$ and $DF = 15$.

To find the scale factor, compare corresponding sides of the triangles: $\text{Scale Factor} = \frac{DF}{AC} = \frac{15}{5} = 3$

Thus, the scale factor by which $\triangle ABC$ is enlarged to get $\triangle DEF$ is 3.

3.5.2 Determining $x$ and $y$

Since the triangles are similar with a scale factor of 3, we can set up proportions to solve for $x$ and $y$.

1. Finding $x$:

   • Corresponding to $AB$ in $\triangle ABC$ and $DE$ in $\triangle DEF$, we use the scale factor.
   • Since the scale factor is 3: $DE = 3 \times AB = 3 \times x$
   • From the diagram, $DE = 12$. $12 = 3 \times x$
   • Solving for $x$: $x = \frac{12}{3} = 4$

   So, $x = 4$.

2. Finding $y$:

   • Corresponding to $BC$ in $\triangle ABC$ and $EF$ in $\triangle DEF$, we use the scale factor.
   • Since $BC = 6$, we have: $y = 3 \times BC = 3 \times 6 = 18$

   So, $y = 18$.

Summary of Answers

• Scale Factor: 3
• Values: $x = 4$ and $y = 18$

Would you like further details or have any questions about these solutions?

Here are five related questions to explore further:

1. How do we determine similarity between two triangles?
2. What are the properties of similar triangles?
3. Can we find the area ratio of two similar triangles given the scale factor?
4. How does the scale factor affect each side length proportionally?
5. What would change in the calculations if the scale factor was different?

Tip: Remember that the ratio of the areas of two similar triangles is the square of the scale factor!