Explain the process of finding the orthocentre of a triangle and also mention the formula.
The orthocentre of a triangle is defined as the point of intersection of all the three altitudes of a triangle.
Answer: Let's understand the process of finding the orthocentre of a triangle and its formula.
Let's understand the concept,
Explanation:
The orthocenter formula helps in locating the coordinates of the orthocenter of a triangle.
Let us consider a triangle PQR, as shown in the figure below.
PA, QB, RC are the perpendicular lines drawn from the three vertices P(\( x_{1}, y_{1})\), Q(\( x_{2}, y_{2})\), and R(\( x_{3}, y_{3})\) respectively of the \( \bigtriangleup \)PQR.
H\( ( x, y) \) is the intersection point of the three altitudes of the triangle.
Step1. Calculate the slope of the sides of the triangle using the formula:
m(slope) = \( \frac{y_{2} \ - \ y_{1}}{x_{2} \ - \ x_{1}} \)
Let slope of PR be given by mPR.
Hence,
mPR = \( \frac{ y_3 \ - \ y_1 }{ x_3 \ - \ x_1 }\)
Similarly,
mQR = \( \frac{ y_3 \ - \ y_2 }{ x_3 \ - \ x_2 }\)
Step2. The slope of the altitudes of the \( \bigtriangleup \)PQR will be perpendicular to the slope of the sides of the triangle.
We know,
\( \begin{align*} \text {Perpendicular slope of line} \ &= \ \frac{-1}{ \text {slope of the line}} \\ &= \frac{-1}{ \text m} \end{align*}\)
The slope of the respective altitudes:
Slope of PA, mPA = \( \frac{-1}{ \text {mQR}}\)
Slope of QB, mQB = \( \frac{ -1}{ \text {mPR}}\)
We will use the slope-point form equation as a straight line to calculate the equations of the lines coinciding with PA and QB.
The generalized equation thus formed by using arbitrary points \( x\) and \( y\) is:
\( \begin{align*} \text {mPA} &= \frac{( y \ - \ y_1 )}{( x \ - \ x_1 )} \\ \text {mQB} &= \frac{( y \ - \ y_2)}{( x \ - \ x_2 )} \end{align*} \)
Thus, solving the two equations for any given values the orthocenter of a triangle can be calculated.