Question

The mid-point of the line segment joining the centroid and the orthocentre of the triangle whose vertices are $(a, b),(a, c)$ and $(d, c)$, is

• A. $\left(\frac{5 a+d}{6}, \frac{b+5 c}{6}\right)$
• B. $\left(\frac{a+5 d}{6}, \frac{5 b+c}{6}\right)$
• C. (a, c)
• D. (0,0)

Answer 1

Answer: (A)

Centroid of $ \Delta A B C=\left(\frac{a+a+d}{3}, \frac{b+c+c}{3}\right)$
$\equiv\left(\frac{2 a+d}{3}, \frac{2 c+b}{3}\right)$
Since, $\Delta A B C$ is right triangle and $\angle B=90^{\circ}$
$\therefore $ Orthocentre of $\Delta A B C$ is $(a, c)$
Mid-point of centroid and orthocentre is
$=\left(\frac{\frac{2 a+d}{3}+a}{2}, \frac{\frac{2 c+b}{3}+c}{2}\right)=\left(\frac{5 a+d}{6}, \frac{5 c+b}{6}\right)$