Question

If in a $△ABC,O$ and $O^{\prime }$ are the incentre and orthocentre respectively, then $(O^{\prime }A+O^{\prime }B+O^{\prime }C)$ is equal to

• A. $2O^{\prime }O$
• B. $O^{\prime }O$
• C. $O{O}^{\prime }$
• D. $2O{O}^{\prime }$

Answer 1

The correct option is A $2O^{\prime }O$

$O^{\prime }A=O^{\prime }O+OA$
$O^{\prime }B=O^{\prime }O+OB$
$O^{\prime }C=O^{\prime }O+OC$
$\Rightarrow O^{\prime }A+O^{\prime }B+O^{\prime }C=3O^{\prime }O+(OA+OB+OC)$ ..... (i)
$\because OA+OB+OC=O{O}^{\prime }=-O^{\prime }O$
$\therefore O^{\prime }A+O^{\prime }B+O^{\prime }C=3O^{\prime }O-O^{\prime }O$ [from Eq. (i)]
$O^{\prime }A+O^{\prime }B+O^{\prime }C=2O^{\prime }O$