Question

In an equilateral triangle, three times the square of one side is equal to four times the square of one of its altitudes.

• A. $True$
• B. $False$

Answer 1

The correct option is A $True$


Let the side of the equilateral triangle be a, and AE be the altitude of $\Delta ABC$.
$\therefore BE=EC=\frac{BC}{2}=\frac{a}{2}$
Applying Pythagoras theorem in $\Delta ABE$, we get
$A{B}^2=A{E}^2+B{E}^2$
$A{B}^2=A{E}^2+{\left(\frac{a}{2}\right)}^2$
$A{E}^2=a^2-\frac{a^2}{4}$
$A{E}^2=\frac{3a^2}{4}$
$4A{E}^2=3a^2$
$\Rightarrow 4\times$ (Square of altitude) = $3\times$ (Square of one side)