Question

Equilateral triangle $DEF$ is inscribed in equilateral triangle $ABC$ such that $D$ is on $BC$. $E$ is on $AC$, $F$ is on $AB$ and $DE\perp BC$, $FD\perp AB$, $FE\perp AC$. Then $\frac{Areaof△DEF}{Areaof△ABC}=$

• A. $\frac{1}{2}$
• B. $\frac{1}{4}$
• C. $\frac{1}{3}$
• D. $\frac{2}{3}$

Answer 1

The correct option is C $\frac{1}{3}$

$x=\frac{a}{\cos {30}^{\circ }}=\frac{2a}{\sqrt{3}};y=\frac{a}{\sqrt{3}}$

$\frac{Areaof△DEF}{Areaof△ABC}=\frac{a^2}{(x+y{)}^2}=\frac{a^2}{(\sqrt{3}a{)}^2}=\frac{1}{3}$