Let's analyze the problem and solve it step by step.

The diagram shows a trapezium $ABCD$ where:

• $AB \parallel CD$,
• $AB = 10 \, \text{cm}$,
• $AD = 18 \, \text{cm}$,
• $BC$ needs to be determined.

The line segments $AE$, $EF$, and $FD$ appear to be equal, meaning the trapezium is divided into congruent parts along its height.

Steps to solve:

1. Identify proportionality: Since $AE$, $EF$, and $FD$ divide $AD$ into three equal parts, we can determine: $AE = EF = FD = \frac{AD}{3} = \frac{18}{3} = 6 \, \text{cm}.$

2. Length of BC: Since $AB \parallel CD$, the segments form a trapezium, and $BC$ corresponds to the third segment after $AB$. To determine the length of $BC$:

   • We add $AE$ and $FD$ to $AB$, as the slant cuts extend symmetrically outward: $BC = AB + 2 \times AE = 10 + 2(6) = 10 + 12 = 22 \, \text{cm}.$

Final Answer:

The measure of $BC$ is 22 cm.

Would you like a step-by-step explanation or further clarification?

Related Questions:

1. What is the height of the trapezium in the given problem?
2. How can you calculate the area of this trapezium?
3. What would happen if $AD$ was divided into 4 equal parts instead?
4. How does the proportionality of segments help in trapeziums?
5. Can you determine $CD$ using the same approach?

Tip:

In trapeziums, parallel lines and proportionality of segments are key to solving for unknown lengths. Use properties of symmetry and equal divisions carefully.