Solution:

Step 1: Prove that $\triangle AEB$ and $\triangle ADC$ are similar.

To prove similarity, we check for equal corresponding angles or proportional sides (AA, SAS, or SSS similarity criteria):

• Angle at $A$:

  • $\angle AEB$ and $\angle ADC$ are vertically opposite angles, so $\angle AEB = \angle ADC$.

• Angle at $B$:

  • $\angle BAE$ is the same as $\angle CAD$, since $AE$ and $AD$ are on the same line.

• Conclusion (AA Similarity Criterion):

  • Since two angles in $\triangle AEB$ and $\triangle ADC$ are equal, the triangles are similar: $\triangle AEB \sim \triangle ADC$

Step 2: Use the proportionality of corresponding sides to calculate $CD$.

From similarity, the corresponding sides are proportional: $\frac{AE}{AD} = \frac{EB}{DC} = \frac{AB}{AC}.$

Let $CD = x$. The total length $AC = AB + BC = 21 + 35 = 56 \, \text{cm}$.

Using the ratios: $\frac{AB}{AC} = \frac{EB}{DC}.$

Substitute the known values: $\frac{21}{56} = \frac{12}{x}.$

Solve for $x$: $\frac{21}{56} = \frac{3}{8}, \quad \frac{3}{8} = \frac{12}{x}.$

Cross-multiply: $3x = 96.$

Solve for $x$: $x = \frac{96}{3} = 32.$

Thus, $CD = 32 \, \text{cm}$.

Final Answer:

The length of $CD$ is $32 \, \text{cm}$.

Would you like further clarification or a more detailed explanation?

Here are 5 related questions to expand understanding:

1. What are the criteria for proving triangle similarity?
2. How do proportionality rules apply in similar triangles?
3. Can you explain how to verify the similarity using SSS or SAS instead of AA?
4. What other geometric properties can help solve triangle similarity problems?
5. How does cross-multiplication work in solving proportionality equations?

Tip: When proving triangle similarity, always ensure at least two corresponding angles are equal or the sides are proportional!