An altitude of a triangle is the perpendicular segment from a vertex to the opposite side of a triangle or to the line containing the opposite side.
Let's draw the altitudes of the vertices of our triangle.
We can see that the altitudes intersect outside the triangle. Therefore, the orthocenter lies outside the triangle. To find its coordinates, we should determine the equations for two of the altitudes and solve the system of these equations. Let's use the altitudes of ST and SR.
Equation of the Altitude of ST
Since ST is horizontal, its altitude will be vertical. From the diagram, we can see that OR is a vertical line through x=- 4. Therefore, the equation of the line for the line segment of the altitude is x=- 4.
Equation of the Altitude of SR
To find the equation for the second altitude, we need the slope of SR. We can use the Slope Formula and the coordinates of S and R to do this.
We found that the slope of SR is - 1. The product of the slopes of two perpendicular lines is -1. This allows us to find the slope of the altitude. Let's call it m_a.
-1* m_a = -1 ⇒ m_a = 1
The slope of the altitude is 1. From the diagram, we also know that the altitude passes through the point T(5,5). We will use point-slope form of a line to write its equation.
y-y_1=m(x-x_1)
Let's substitute 1 for m and ( 5, 5 ) for (x_1,y_1) in the formula.
This final equation is the equation of the line for the line segment of the altitude.
Solving for the Coordinates
Finally, we can solve the system of the found equations to find the coordinates of their intersection.
x=-4 y=x ⇔ x=-4 y=-4
Therefore, the coordinates of the orthocenter are (- 4,- 4).
