Question : In $\triangle \text{ABC}, \mathrm{DE} \| \mathrm{BC}$ and $\frac{\text{AD}}{\text{DB}}=\frac{4}{5}$. If $\mathrm{DE}=12 \mathrm{~cm}$, find the length of $\mathrm{BC}$.
Option 1: 48 cm
Option 2: 12 cm
Option 3: 30 cm
Option 4: 27 cm
Correct Answer: 27 cm
Solution :
According to the question,
$\mathrm{DE} \| \mathrm{BC}$
$\frac{\text{AD}}{\text{DB}}=\frac{4}{5}$
DE = 12 cm
By the SAS criterion of similarity, we have
△ABC $\sim$ △ADE
$\therefore$ $\frac{\text{AB}}{\text{AD}} = \frac{\text{BC}}{\text{DE}}$
⇒ $\text{BC} = \frac{\text{AB}}{\text{AD}}\times \text{DE}$
⇒ $\text{BC} = \frac{\text{AD+DB}}{\text{AD}}\times 12$
⇒ $\text{BC} = (1+\frac{\text{DB}}{\text{AD}})\times 12$
⇒ $\text{BC}=12\times(1+\frac{5}{4}) = 3\times 9 = 27$ cm
Hence, the correct answer is 27 cm.
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