Question

Explain the process of finding the orthocenter of a triangle and also mention the formula.

Answer 1

Step 1: Definition

1. The orthocenter of a triangle is the point where the perpendicular drawn to the sides of the triangle from opposite vertices, intersect each other.
2. The orthocenter lies inside the triangle, For an acute angle triangle.
3. The orthocenter lies outside the triangle, For the obtuse angle triangle.
4. The orthocenter lies on the vertex of the right angle, For a right angle triangle.

Step 2: Find the orthocenter of a triangle

1. Let ABC be a triangle.
2. Here AD, BE, and CF are the perpendiculars drawn from the vertices A(x₁,y₁), B(x₂,y₂), and C(x₃,y₃), respectively
3. O is the point of intersection of the three altitudes.

First, we will find the slope of the sides of the triangle, using the formula :

$\Rightarrow m_{AB}=\frac{(y_2-y_1)}{(x_2-x_1)}$

Then, the slope of altitude will be :

$\Rightarrow m_{CF}=\frac{-(x_2-x_1)}{\left(y_2-y_1\right)}$ $\left[\mathrm{slope}\mathrm{of}\mathrm{perpendicular}=\frac{-1}{\mathrm{slope}\mathrm{of}\mathrm{respective}\mathrm{side}}\right]$

Step 3: Further simplification

Therefore, $m_{CF}=\frac{-1}{m_{AB}}$

Similarly,

$$\begin{gathered} m_{BE}=\frac{-1}{m_{AC}} \\ {m}_{AD}=\frac{-1}{m_{BC}} \end{gathered}$$

Now, the equation of AD, BE and CF from slope point form of the equation of a line.

$$\begin{gathered} \Rightarrow m_{BE}=\frac{\left(y-y_2\right)}{\left(x-x_2\right)}.....\left(1\right) \\ \Rightarrow m_{AD}=\frac{\left(y-y_1\right)}{\left(x-x_1\right)}......\left(2\right) \\ \Rightarrow m_{CF}=\frac{(y-y_3)}{\left(x-x_3\right)}.......\left(3\right) \end{gathered}$$

Hence, from the above three equations take any two equations and solve for the value of x and y

Hence, x and y are required coordinates of the orthocenter.