To solve the problem of finding the length of line $CD$, we apply the principles of similar triangles. From the diagram:

Given:

• $AB = 12 \, \text{cm}$,
• $AC = 18 \, \text{cm}$,
• $BE = 10 \, \text{cm}$.

We observe that:

• $\triangle ABE \sim \triangle ACD$ (by the AA similarity criterion since the angles are shared and corresponding sides are proportional).

Step 1: Set up the ratio

Using the similarity property, we know: $\frac{AB}{AC} = \frac{BE}{CD}.$

Substituting the known values: $\frac{12}{18} = \frac{10}{CD}.$

Step 2: Simplify the ratio

$\frac{12}{18} = \frac{2}{3}.$ So, we rewrite the equation as: $\frac{2}{3} = \frac{10}{CD}.$

Step 3: Solve for $CD$

Cross-multiply to find $CD$: $2 \cdot CD = 3 \cdot 10.$ $2 \cdot CD = 30.$ $CD = \frac{30}{2} = 15 \, \text{cm}.$

Final Answer:

$CD = 15 \, \text{cm}.$

Let me know if you'd like a step-by-step breakdown of the solution, and feel free to ask any clarifications!

Related Questions:

1. How can you verify that two triangles are similar?
2. What are the proportionality properties of similar triangles?
3. How can you find an unknown side using similar triangles?
4. What is the importance of cross-multiplication in solving ratios?
5. How can scaling affect the dimensions of a triangle while maintaining similarity?

Tip:

Always confirm the similarity of triangles