Question

If $E$ is a point on the side $CA$ of an equilateral triangle $ABC,$ such that $BE\perp CA,$ then $A{B}^2+B{C}^2+C{A}^2=$

• A. $2B{E}^2$
• B. $3B{E}^2$
• C. $4B{E}^2$
• D. $6B{E}^2$

Answer 1

The correct option is C $4B{E}^2$

Given: $ABC$ is an equilateral triangle and $BE\perp CA$
Since, $ABC$ is an equilateral triangle

So, $E$ is the mid point of $AC$
Now, In $△BEC$
$B{E}^2+C{E}^2=B{C}^2$
$B{E}^2+{\left(\frac{BC}{2}\right)}^2=B{C}^2$

$B{E}^2=\frac{3}{4}B{C}^2$ ....... $\left(1\right)$

Now, $A{B}^2+B{C}^2+C{A}^2=3B{C}^2$ ...... $\left(2\right)$
From $\left(1\right)$ and $\left(2\right),$ we get
$A{B}^2+B{C}^2+C{A}^2=4B{E}^2$