In triangle ABC, points D and E are on AB and AC respectively such that DE is parallel to BC. If $$AD = 6 \, ext{cm}$$, $$DB = 4 \, ext{cm}$$, $$AE = 9 \, ext{cm}$$, then the length of $$EC$$ (in cm) is:

Question

In triangle ABC, points D and E are on AB and AC respectively such that DE is parallel to BC. If $$AD = 6 \, 	ext{cm}$$, $$DB = 4 \, 	ext{cm}$$, $$AE = 9 \, 	ext{cm}$$, then the length of $$EC$$ (in cm) is:

Accepted Answer

1. **Apply the similarity property of triangles:**  
   - Since DE is parallel to BC, triangles $$	riangle ADE$$ and $$	riangle ABC$$ are similar by the Basic Proportionality Theorem (Thales' theorem).

2. **Set up the ratio using the sides of similar triangles:**  
   - $$\frac{AD}{AB} = \frac{AE}{AC}$$.

3. **Calculate AB and AC:**  
   - $$AB = AD + DB = 6 + 4 = 10 \, 	ext{cm}$$.  
   - $$AC = AE + EC$$ (let EC = x).

4. **Substitute values into the ratio equation:**  
   - $$\frac{6}{10} = \frac{9}{9 + x}$$.

5. **Solve for x (EC):**  
   - Cross-multiply:  
     $$
     6(9 + x) = 90.
     $$  
   - Simplify:  
     $$
     54 + 6x = 90 \implies 6x = 36 \implies x = 6.4.
     $$

Thus, the length of $$EC$$ is **6.4 cm**.

Answer

Answer

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