Question

Let \[\overrightarrow p \] be the position vector of the orthocentre and \[\overrightarrow g \] be the position vector of the centroid of the triangle ABC where circumcentre is the origin. If \[\overrightarrow p = K\overrightarrow g \], then what is the value of K?
(a). 3
(b). 2
(c). \[\dfrac{1}{3}\]
(d). \[\dfrac{2}{3}\]

Answer 1

Hint: Recall the definitions for the orthocentre, centroid, and the circumcentre of a triangle. The centroid divides the distance from the orthocentre to the circumcentre in the ratio 2:1.

Complete step by step answer:
Use this theorem to find the value of K.

The orthocentre is the point of intersection of the three altitudes of the triangle drawn from the vertices to the opposite sides.

The centroid is the point of intersection of three medians of the triangle. A median is a straight line that joins the vertex to the midpoint of the opposite side.

The circumcentre is the point of the intersection of the three perpendicular bisectors of the sides of a triangle. It is the center of the circle circumscribed by the triangle.

The orthocentre, the centroid, and the circumcentre of any triangle are collinear. The centroid divided the distance from the orthocentre to the circumcentre in the ratio 2:1. The line on which these three points lie is called the Euler line of the triangle.

In this problem, the circumcentre is the origin. The distance of orthocentre from circumcentre is three times the distance of the centroid from circumcentre and the direction is the same. Then, we have:

\[\overrightarrow p = 3\overrightarrow g \]

Hence, the value of K is 3.

Hence, the correct answer is option (a).

Note: The option (d) is also very likely to choose because if orthocentre’s position vector and the centroid’s position vector are interchanged, then this is the answer but in this question, it is a wrong answer.