You're correct; I made an error by assuming the value of $AB$. Let's recalculate part (b) without that assumption.

(b) Calculate the length of $AC$:

Given:

• $BC = 6$ cm
• $AD = 4$ cm
• $\angle ABC = \angle ACD$

From part (a), $\triangle ABC \sim \triangle ACD$, so the corresponding sides are proportional: $\frac{AB}{AC} = \frac{AC}{AD}$

Let $AC = x$. Then the proportion becomes: $\frac{AB}{x} = \frac{x}{4}$

We also know: $\frac{BC}{CD} = \frac{AC}{AD} = \frac{x}{4}$ Since $BC = 6$ cm and $\frac{BC}{CD} = \frac{x}{4}$, we find $CD$: $CD = \frac{BC \times AD}{x} = \frac{6 \times 4}{x} = \frac{24}{x}$

Thus, the length $AC$ would require more information or another method (such as solving for the value of $AB$ directly) to be determined precisely. Given the information, without $AB$, $AC$ cannot be determined directly using the standard properties of similar triangles alone.

Since there isn't enough information provided, the answer for part (b) can't be calculated directly without making additional assumptions or having more details.

Answer (b): Insufficient data to determine $AC$

Would you like to ask any other question or need further clarification? Here are 5 related questions:

1. What are some methods for finding the length of a side in a triangle using the Law of Sines?
2. How does the Law of Cosines help in finding the missing side of a triangle?
3. What additional information could help determine the length of $AC$ here?
4. How can you verify the similarity of triangles using different properties?
5. What is the significance of knowing the length of a side in similar triangles for solving geometric problems?

Tip: When dealing with similar triangles, make sure all necessary side lengths or angle measures are given or deducible to solve the problem correctly.