Question

The coordinates of the orthocenter of the triangle formed by $(0,0),(8,0)$ and $(4,6)$ is

• A. $(4,0)$
• B. $(6,3)$
• C. $(6,0)$
• D. None of these

Answer 1

The correct option is D

None of these

Explanation for the correct option:

Step 1: Finding the slope of AD and BC

The vertices of the triangle are:

$A(0,0)$,$B(8,0)$ and $C(4,6)$.

$D$ is point on $BC$ such that $AD\perp BC$

Now, the slope of $BC$$=\frac{y_2-y_1}{x_2-x_1}$

$=\frac{6-0}{4-8}$

$=-\frac{3}{2}$

Product of slope of two perpendicular segments is $-1$.

$AD\perp BC$

$\therefore \text{Slope of}AD=\frac{-1}{\text{Slope of}BC}$

. $$\begin{gathered} =\frac{-1}{\frac{-3}{2}} \\ =\frac{2}{3} \end{gathered}$$

Step 2: Finding the equation of $AD$

Equation of a line in slope-point form is

$y-y_1=m\left(x-x_1\right)$

Hence, the equation of $AD$:

$$\begin{gathered} y-0=\frac{2}{3}\left(x-0\right) \\ \Rightarrow 3y=2x \\ \Rightarrow 3y-2x=0\to \left(i\right) \end{gathered}$$

Step 3: Finding the slope of $AC$ and $BE$

$E$ is point on $AC$ such that $AC\perp BE$.

Slope of $AC$$=\frac{6-0}{4-0}$

$=\frac{3}{2}$

Product of slope of two perpendicular segments is $-1$.

Therefore, the slope of $BE$$=\frac{-2}{3}$

Step 4: Finding the equation of BE

Equation of line in slope-point form is

$y-y_1=m\left(x-x_1\right)$

Equation of $BE$

$$\begin{gathered} y-0=-\frac{2}{3}\left(x-8\right) \\ \Rightarrow 3y=-2x+16 \\ \Rightarrow 3y+2x=16\to \left(ii\right) \end{gathered}$$

Step 5: Find the coordinates of the orthocentre

Since The orthocenter is the point of intersection of the attitudes.

Solving $\left(i\right)$ and $\left(ii\right)$ we will get the value $O$ which is the orthocenter of the triangle.

Add equation $\left(i\right)$ and $\left(ii\right)$

$$\begin{gathered} \Rightarrow 6y=16 \\ y=\frac{16}{6} \\ =\frac{8}{3} \end{gathered}$$

Substituting the value in the equation $\left(i\right)$

$\begin{array}{rcl}3\times \left(\frac{8}{3}\right)-2x& =& 0\\ & \Rightarrow & 8=2x\\ x& =& 4\end{array}$

Therefore, the coordinates of the orthocenter are $\left(4,\frac{8}{3}\right)$

Hence, option (D) is the correct answer