Question

The coordinates of the vertices of a triangle are $(0,1),(2,3)$ and $(3,5)$. Find orthocentre of the triangle.

• A. $(14,-6)$
• B. $(2,6)$
• C. $(\frac{-9}{2},\frac{15}{2})$
• D. $(\frac{9}{2},3)$

Answer 1

The correct option is A $(14,-6)$

Let vertices of triangle be $A(0,1),B(2,3)$ and $C(3,5)$

And coordinate of orthocentre be $H(x,y)$

Now slope of $AB=\frac{3-1}{2-0}=1$

Then slope of line perpendicular to AB is $-1$

and mid point of AB is $E(1,2)$

But slope of $HE=\frac{y-2}{x-1}$

Equating it with slope of line perpendicular to AB we get

$\frac{y-2}{x-1}=-1\Rightarrow y-2=-x+1\Rightarrow x+y=1$ ...(1)

Similarly from BC

$\frac{y-4}{x-\frac{5}{2}}=-\frac{1}{2}\Rightarrow 2y-8=-x+\frac{5}{2}\Rightarrow 4y-16=-2x+5\Rightarrow 2x-4y=21$ ...(2)

Solving (1) and (2) we get

$(x,y)=(14,-6)$