To calculate the length of $AC$ using the scale factor, let’s proceed systematically.

Step 1: Identify corresponding triangles

Given:

• $\triangle ABC$ and $\triangle ADE$
• $BC \parallel DF$, which means that the triangles are similar (since corresponding angles are equal).
• This similarity implies that the sides are proportional.

Step 2: Set up the proportion

The corresponding sides are:

• $AB$ corresponds to $AD$
• $BC$ corresponds to $DF$
• $AC$ corresponds to $AE$

The scale factor between the two triangles will be the ratio of corresponding sides: $\text{Scale Factor} = \frac{BC}{DF} = \frac{6}{10} = 0.6$

Step 3: Apply the scale factor to find $AC$

Since the triangles are similar, the ratio of any two corresponding sides must be equal to the scale factor.
So, we can write the relation between $AC$ and $AE$ as: $\frac{AC}{AE} = \frac{BC}{DF} = 0.6$

Now, let's express $AE$ in terms of $AC$. Since $AE$ is longer than $AC$, it will be: $AE = AC \times \frac{DF}{BC} = AC \times \frac{10}{6}$

However, you haven’t provided the exact measurement of either $AC$ or $AE$. If you need to calculate only $AC$ or verify another unknown, please provide the missing detail, or I can help you further with another approach.

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Would you like me to explore further possibilities? For example, are you also interested in solving for $AD$?

Here are five related questions you may ask next:

1. How do we use the Pythagorean theorem for similar triangles?
2. What are the criteria for two triangles to be similar?
3. If BC = 6 cm and DF = 12 cm, what would be the new length of AC?
4. How do scale factors affect area and perimeter?
5. How can proportionality help in real-life applications?

Tip: Always ensure the sides you compare are truly corresponding sides when working with similar triangles!