Question

$ABC$ is a triangle right-angled at $B$ and $D$ is a point on $BC$ produced $(BD>BC),$ such that $BD=2DC$. Which one of the following is correct$?$

• A. $A{C}^2=A{D}^2-3C{D}^2$
• B. $A{C}^2=A{D}^2-2C{D}^2$
• C. $A{C}^2=A{D}^2-4C{D}^2$
• D. $A{C}^2=A{D}^2-5C{D}^2$

Answer 1

The correct option is A $A{C}^2=A{D}^2-3C{D}^2$

Given, $BD=2DC$
$\Rightarrow$ $BC+CD=2DC\Rightarrow BC=DC$
In $△ABC$, applying pythagoras theorem
$A{C}^2=A{B}^2+B{C}^2$ ....(i)
In $△ABD$, appying pythagoras theorem
$A{D}^2=A{B}^2+B{D}^2$ .....(ii)

Subtracting equations (i) and (ii), we have
$A{C}^2-A{D}^2=B{C}^2-B{D}^2$
$\Rightarrow$ $A{C}^2-A{D}^2=C{D}^2-{\left(2CD\right)}^2$
$=C{D}^2-4C{D}^2$

$=-3C{D}^2$
$\Rightarrow$ $A{C}^2=A{D}^2-3C{D}^2$