Question

Find the orthocenter of the triangle whose vertices are (5,2), (3,7) and (4,9)?

Answer 1

To answer this question, we must first understand that the orthocenter is nothing more than the point of intersection of the triangle's three elevations. So, to get the point of intersection of the altitudes, we will first discover the slope of the triangle's altitudes, then the equation of the altitudes, and last their point of junction.

Complete step by step solution:
The orthocenter is the point where a triangle's elevations meet. A line segment that passes through the vertex of a triangle and is perpendicular to the opposing side is known as an altitude.
This is the orthocenter if you find the intersection of any two of the three heights because the third altitude will also cross the others at this place.

We name the vertices in triangle ABC as $A = (5,2)$ , $B = (3,7)$ , $C = (4,9)$
We will now find the slope of the AB using the slope formula $m = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
$\Rightarrow {m_{AB}} = \dfrac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}$
$\Rightarrow {m_{AB}} = \dfrac{{7 - 2}}{{3 - 5}} = - \dfrac{5}{2}$
We know that the slope of the line segment perpendicular to a line is the opposite sign of reciprocity of the slope of the line.
The slope of line perpendicular to the line having slope $- \dfrac{5}{2}$ is $\dfrac{2}{5}$
We will use the point slope formula $\;y - {y_1} = m(x - {x_1})$ to find the equation of the altitude from vertex C to side AB.
We will find the equation of line
$\Rightarrow \;y - 9 = \dfrac{2}{5}(x - 4)$
$\Rightarrow \;y - 9 = \dfrac{2}{5}x - \dfrac{8}{5}$
$\Rightarrow \;y = \dfrac{2}{5}x + \dfrac{{37}}{5}$ (1)
We will determine the slope of one of the triangle's other sides to determine the equation for a second altitude.
We will find the slope of the line BC
$\Rightarrow {m_{BC}} = \dfrac{{9 - 7}}{{4 - 3}} = 2$
We will find the slope of the line perpendicular to the line having slope 2 is $- \dfrac{1}{2}$
We will find the equation of the altitude from vertex A to the side BC
$\Rightarrow y - 2 = - \dfrac{1}{2}(x - 5)$
$\Rightarrow y - 2 = - \dfrac{1}{2}x + \dfrac{5}{2}$
$\Rightarrow y = - \dfrac{1}{2}x + \dfrac{9}{2}$ (2)
We will now find the point of intersection of the equation 1 and 2
We will use substitution method to find the solution of the equations
When we solve the equation, we get the value of x and y as $- \dfrac{{29}}{9}$ and $\dfrac{{55}}{9}$
Hence, the orthocentre of the triangle having vertices are (5,2), (3,7) and (4,9) is $\left( { - \dfrac{{29}}{9},\dfrac{{55}}{9}} \right)$.

Note:
We have a slope and a point, we can use the slope point form to find an equation. It's important to remember that the slope of a perpendicular line is the reciprocal of the slope of the provided line.