Question
In the given figure, $\angle\text{ABC}=90^\circ$ and $\text{BD}\perp\text{AC}.$ If $AB = 5.7\ cm$, $BD = 3.8\ cm$, and $CD = 5.4\ cm$ find $BC.$
✓
Answer
Given that $AB = 5.7\ cm, BD = 3.8\ cm$, and $CD = 5.4\ cm$
In $\triangle\text{CBA}$ and $\triangle\text{CDB}$
$\angle\text{CBA}=\angle\text{CDB}=90^\circ$
$\angle\text{C}=\angle\text{C}$ (common)
Therefore, $\triangle\text{CBA}\sim\triangle\text{CDB}$ (by $AA$ similarities)
$\Rightarrow\frac{\text{BC}}{\text{CD}}=\frac{\text{BA}}{\text{BD}}$
$\Rightarrow\frac{\text{BC}}{\text{5.4}}=\frac{\text{5.7}}{\text{3.8}}$
$\Rightarrow\text{BC}=\frac{\text{5.7}\times5.4}{\text{3.8}}$
$\therefore\text{BC}=8.1\text{cm}$
Hence, $BC= 8.1\ cm$
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