Question

In an equilateral $△ABC$, if $AD\perp BC$ then prove that $A{D}^2=3B{D}^2$

Answer 1

Let each side be a unit

then height $AD=\frac{\sqrt{3}}{2}aA{D}^2=(\frac{\sqrt{3}}{2}a{)}^2=\frac{3}{4}{a}^2$

and $B{D}^2=\frac{a^2}{4}$

$\Rightarrow \frac{A{D}^2}{B{D}^2}=\frac{3a^2}{4}÷\frac{a^2}{4}=3\Rightarrow A{D}^2=3B{D}^2$