In $\triangle ABC$, $DE \parallel BC$ (as shown in the figure). If $AD = 4$ cm, $AB = 9$ cm and $AC = 13.5$ cm, then the length of $EC$ is :
Generated by claude-sonnet-4-6 · 2026-06-15 10:23 · grounding rag
Model Answer
By Basic Proportionality Theorem (DE ∥ BC):
$$\frac{AD}{DB} = \frac{AE}{EC}$$
DB = AB − AD = 9 − 4 = 5 cm
$$\frac{4}{5} = \frac{AE}{EC}$$
Also, AE = AC − EC = 13.5 − EC
$$\frac{4}{5} = \frac{13.5 - EC}{EC}$$
$$4 \cdot EC = 5(13.5 - EC) = 67.5 - 5\,EC$$
$$9\,EC = 67.5 \implies EC = 7.5 \text{ cm}$$
Answer: (B) 7.5 cm
Source: Chapter 6, Section 6.3 (Theorem 6.1 — Basic Proportionality Theorem)
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Explanation
- The key result is Theorem 6.1 (BPT): if DE ∥ BC, then AD/DB = AE/EC.
- Find DB first (AB − AD), then set up the ratio with AE = AC − EC to solve for EC.
- Common mistake: using AD/AB instead of AD/DB — always use the segment ratios, not the whole-side ratios, unless the formula is rearranged as in Example 1.