Discussion Forum

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

Answer:

Explanation:

Key Idea    Orthocentre, centroid and circumcentre are collinear and centroid divide orthocentre and circumcentre in 2:1 ratio.

We have orthocentre and centroid of a triangle be A ( -3,5) and ( 3,3) respectively and  C circumcentre

Clearly , $AB= \sqrt{(3+3)^{2}+( 3-5)^{2} }=\sqrt{36+4}=2\sqrt{10}$

We know that, AB:BC=2:1

$\therefore$       BC= $\sqrt{10}$

Now,

AC=AB+BC=$2\sqrt{10}+\sqrt{10}=3\sqrt{10}$

Since, AC is a diameter of circle

$\therefore$    $r=\frac{AC}{2}$

$\Rightarrow$      $r=\frac{3\sqrt{10}}{2}=3\sqrt{\frac{5}{2}}$