Question

Let $A,B,C,D$ be four points in a plane with position vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}$ respectively such that $(\overrightarrow{a}-\overrightarrow{d}).(\overrightarrow{b}-\overrightarrow{c})=(\overrightarrow{b}-\overrightarrow{d}).(\overrightarrow{c}-\overrightarrow{a})=0$.Show that the point $D$ is the orthocentre of $△ABC$.

• A. in centre
• B. circumcentre
• C. orthocentre
• D. centroid

Answer 1

The correct option is A in centre

$(\overrightarrow{a}-\overrightarrow{d}).(\overrightarrow{b}-\overrightarrow{c})=0$
$\Rightarrow AD\perp BC$
$(\overrightarrow{b}-\overrightarrow{d}).(\overrightarrow{c}-\overrightarrow{a})=0$
$\Rightarrow BD\perp AC$
$\therefore AD,BD$ are the two altitudes and $D$ is the orthocentre.