Question

If a triangle $ABC$has vertices $A(-1,7),B(-7,1)$, and$C(5,-5)$, then its orthocentre has coordinates

• A. $\left(-3,3\right)$
• B. $\left(-3/5,3/5\right)$
• C. $\left(3/5,-3/5\right)$
• D. $(3,-3)$

Answer 1

The correct option is A

$\left(-3,3\right)$

Explanation for the correct option:

Step1. Draw the triangle :

Step2. Find the ortho center of the triangle:

Let $H\left(x,y\right)$ be ortho center of the triangle and let $AD$and$BE$ be lines passing through the orthocenter.

Thus

$$\begin{gathered} m_{AH}·m_{BC}=-1 \\ \Rightarrow \left(\frac{y-7}{x+1}\right)\left(\frac{-5-1}{5+7}\right)=-1 \\ \Rightarrow 2x-y+9=0.........\left(i\right) \end{gathered}$$

and

$$\begin{gathered} m_{BH}·m_{AC}=-1 \\ \Rightarrow \left(\frac{y-1}{x+7}\right)\left(\frac{7-5}{-1-5}\right)=-1 \\ \Rightarrow x-2y+9=0.......\left(ii\right) \end{gathered}$$

Solving equations (i) and (ii) we get

$(x,y)=(-3,3)$

Hence, the correct option is (A).