Question

Write the coordinates of the circumcentre of a triangle whose centroid and orthoentre are at (3,3) and (-3,5) respectively.

Answer 1

$\text{We know that the orthocentre,centroid and the circumcentre of a triangle are collinear,}\text{Also we know that the distance between orthocentre and centroid is twice the distance between centroid and the circumcentre,}\text{Given that centroid of a triangle is G(3,3),orthocentre is H(-3,5)}\text{Let C(x,y) be the coordinates of the circumcentre of the triangle .}$

$\text{Hence using section formula,we have,}\frac{2\times x-3\times 1}{2+1}=3and\frac{2\times y+5\times 1}{2+1}\Rightarrow \frac{2x-3}{3}=3and\frac{2y+5}{3}=3\Rightarrow 2x-3=9and2y+5=9\Rightarrow 2x=9+3and2y=9-5\Rightarrow 2x=12and2y=4\Rightarrow x=6andy=2\text{Thus,the cirumcentre of the triangle is C(6,2)}$