Question
Δ ABC and ΔDEF are equilateral triangles. If A(ΔABC) : A(ΔDEF) = 1 : 2 and AB = 4, find DE.
✓
Answer
We know that, all the angles of an equilateral triangles are equal, i.e., 60°.⇒ Δ ABC~Δ DEF ……(By AAA Similarity Test)
$\Rightarrow \frac{ A (\triangle ABC )}{ A (\triangle DEF )}=\frac{ AB ^2}{ DE ^2}$
And, $\frac{ A (\triangle ABC )}{ A (\triangle DEF )}=\frac{1}{2}$ (Given)
$\Rightarrow \frac{ AB ^2}{ DE ^2}=\frac{1}{2}$
$\Rightarrow D E^2=2 \times 4^2(\because A B=4)$
$\Rightarrow D E=\sqrt{ } 32$
$\Rightarrow D E=4 \sqrt{ } 2$
$\Rightarrow \frac{ A (\triangle ABC )}{ A (\triangle DEF )}=\frac{ AB ^2}{ DE ^2}$
And, $\frac{ A (\triangle ABC )}{ A (\triangle DEF )}=\frac{1}{2}$ (Given)
$\Rightarrow \frac{ AB ^2}{ DE ^2}=\frac{1}{2}$
$\Rightarrow D E^2=2 \times 4^2(\because A B=4)$
$\Rightarrow D E=\sqrt{ } 32$
$\Rightarrow D E=4 \sqrt{ } 2$
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