Question

In an equilateral triangle $ABC$ , if $AD$ is perpendicular to $BC$ , then

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Since, $△ABC$ is equilateral, therefore, $AB=BC=CA$

Since, $AD$ is perpendicular to $BC$, therefore, $BD=\frac{1}{2}AB,DC=\frac{1}{2}AB$

In $△ABD$

$A{B}^2=A{D}^2+B{D}^2$ --------(1)

In $△ACD$

$A{C}^2=A{D}^2+D{C}^2$ ---------(2)

Adding (1) and (2), we get

$A{B}^2+A{C}^2=2A{D}^2+\frac{1}{4}A{B}^2+\frac{1}{4}A{B}^2$

$\Rightarrow 2A{B}^2=2A{D}^2+\frac{1}{2}A{B}^2$ [since $AC=AB$]

$\Rightarrow (2-\frac{1}{2})A{B}^2=2A{D}^2$

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

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