Question

If ABC be an equilateral triangle and $AD\perp BC$. Then ${AB}^2+{BC}^2+{CA}^2=?$

• A. $3{AD}^2$
• B. $5{AD}^2$
• C. $2{AD}^2$
• D. None of these

Answer 1

The correct option is D None of these

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

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

$A{B}^2+A{C}^2=2A{D}^2+B{D}^2+C{D}^2$

$A{B}^2+A{C}^2+B{C}^2=2A{D}^2+a^2+\frac{a^2}{4}+\frac{a^2}{4}=4A{D}^2$ $(a^2-\frac{a^2}{4}=A{D}^2)$