Question

Assertion (A) If $(-1,3,2)$ and $(5,3,2)$ are respectively the orthocentre and circumcentre of a triangle, then $(3,3,2)$ is its centroid.
Reason (R) Centroid of the triangle divides the line segment joining the orthocentre and the circumcentre in the ratio $1:2$.
Which one of the following is true?

• A. (A) and (R) are true and (R) is the correct explanation to (A)
• B. (A) and (R) are true, but (R) is not the correct explanation to (A)
• C. (A) is true, (R) is false
• D. (A) is false, $(R)$ is true

Answer 1

Answer: (C)

Except the equilateral triangle, the centroid, orthocentre and circumcentre are collinear and centroid divides the line segment joining the orthocentre and the circumcentre in the ratio $2:1$. So, if $(-1,3,2)$ and $(5,3,2)$ are respectively the orthocentre and circumcentre of triangle, then coordinate of centroid is
$\left(\begin{array}{l}\frac{(-1\times 1)+(5\times 2)}{1+2},\frac{(3\times 1)+(3\times 2)}{1+2}\\ \frac{(2\times 1)+(2\times 2)}{1+2}\end{array}\right)=(3,3,2)$
So, $(A)$ is true and $(R)$ is false.