The problem presents a triangle $\triangle ACD$ with the following details:

• $B$ is a point on $AC$.
• $E$ is a point on $AD$.
• $BE \parallel CD$.

Given measurements are:

• $AE = 4$ cm
• $AC = 11.7$ cm
• $BE = 6$ cm
• $CD = 13.5$ cm

To solve part (a): Calculate the length of $AB$.

Since $BE \parallel CD$, triangles $\triangle ABE$ and $\triangle ACD$ are similar by the Basic Proportionality Theorem (or Thales' theorem). This means that the sides of the triangles are proportional:

$\frac{AB}{AC} = \frac{AE}{AD}$

We need to find $AB$, so let's first find $AD$.

Step 1: Find $AD$.

Since $BE \parallel CD$, the proportion also holds for the line segments $\frac{BE}{CD}$:

$\frac{BE}{CD} = \frac{AE}{AD}$

Substitute the known values:

$\frac{6}{13.5} = \frac{4}{AD}$

Cross-multiplying gives:

$6 \cdot AD = 4 \cdot 13.5$

$6 \cdot AD = 54$

$AD = 9 \ \text{cm}$

Step 2: Find $AB$.

Now use the similarity ratio to find $AB$:

$\frac{AB}{11.7} = \frac{4}{9}$

Cross-multiplying gives:

$AB = \frac{4}{9} \cdot 11.7$

$AB = 5.2 \ \text{cm}$

So, the length of $AB$ is 5.2 cm.

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. How can we verify the similarity of triangles using parallel lines?
2. What is the Basic Proportionality Theorem and how is it applied?
3. How do we determine the ratio of sides in similar triangles?
4. What other methods could we use to find the length of $AB$?
5. How does the parallel line $BE$ impact the proportions of triangle $ACD$?

Tip: When working with similar triangles, always check for parallel lines and corresponding sides, as they can simplify many calculations.