Let the orthocentre and centroid of a triangle be A(-3,5) and B(3,3) respectively.

Question

Let the orthocentre and centroid of a triangle be $A(-3,5)$ and $B(3,3)$ respectively. If $C$ is the circumcentre of this triangle, then the radius of the circle having line segment $AC$ as diameter, is:

Accepted Answer

In a triangle, orthocentre, centroid and circumcenter are collinear and centroid divides orthocenter and circumcenter in 2 : 1 ratio.

Here AB = 2 BC

$$AB = \sqrt {{{\left( {3 + 3} \right)}^2} + {{\left( {3 - 5} \right)}^2}} $$

$$ = \sqrt {36 + 4} $$

$$ = \sqrt {40} $$

$$ = 2\sqrt {10} $$

$$\therefore\,\,\,\,$$  BC $$ = \sqrt {10} $$

$$\therefore\,\,\,\,$$ AB + BC

$$ = 2\sqrt {10}  + \sqrt {10} $$

$$ = 3\sqrt {10} $$

AC is the diameter of the circle.

$$\therefore\,\,\,$$  Radius $$ = {1 \over 2}\,\,$$ AC

$$ = {1 \over 2} \times 3\sqrt {10} $$

$$ = 3\sqrt {{5 \over 2}} $$

Solutions · 1