Question

Two coordinate of the orthocentre of the triangle that the coordinates of mid-points of its sides as $(0,0)(1,2)$ and $(-6,3)$ is:

• A. $(7,0)$
• B. $(-4,5)$
• C. $(-5,5)$
• D. $(-4,4)$

Answer 1

The correct option is A $(7,0)$

By using mid point formula

$\begin{array}{c}\Rightarrow \frac{x_1+x_2}{2}=0,x_1+x_2=0\to \left(i\right)\\ \Rightarrow \frac{y_2+y_2}{2}=0,y_1+y_2=0\to \left(ii\right)\\ \Rightarrow \frac{x_2+x_3}{2}=1,x_2+x_3=2\to \left(iii\right)\\ \Rightarrow \frac{y_2+y_3}{2}=2,y_2+y_3=4\to \left(iv\right)\\ \Rightarrow \frac{x_3=x_1}{2}=-6,x_3+x_1=-12\to \left(v\right)\\ \Rightarrow \frac{y_1+y_3}{2}=3,y_1+y_3=6\to \left(vi\right)\end{array}$

$\begin{array}{c}Adding\left(i\right)and\left(iii\right),weget2x_2+x_1+x_3=2\\ 2x_2-12=2,x_2=7,x_1=-7,x_3=-19\end{array}$

$\begin{array}{c}Adding\left(ii\right)and\left(iv\right),weget2y_2+y_1+y_3=4\\ \Rightarrow 2y_2=4-6,y_2=\frac{2}{2}=-1,y_1=1,y_3=5\end{array}$

Then ortho centre is $(7,0)$ because two coordinates are alternate.