Answer:
Question: What is the orthocenter of a triangle with the vertices (-1,2) (5,2) and (2,1)?
The coordinates of point A are (-1,2), point B are (5,2), and point C are (2,1).
The orthocent is the intersection of the three altitudes. An altitude goes from a vertex and is perpendicular to the line containing the opposite side.
In the coordinate plane the equations of the altitudes can be found and then a system of equations can be solved.
Altitude 1. From point C perpendicular to the line containing side AB.
Slope of line AB is 0 (horizontal line), a vertical line is perpendicular to a horizontal line. Thus, the equation of altitude 1 is x=2 .
Altitude 2. From point B perpendicular to the line containing side AC.
Slope of line AC is −13 , the slope of a line perpendicular to line AC is 3. The equation of altitude 2 is y=3x−13
Altitude 3. From point A perpendicular to the line containing side BC.
Slope of line BC is 13 , the slope of a line perpendicular to line BC is −3 . The equation of altitude 3 is y=−3x−1
The orthocenter is the point where all three altitudes intersect.
x=2
y=3x−13
y=−3x−1
Use substitution to solve the first two equations y=3(2)−13=−7
The orthocenter is the point (2,−7)
we did not need the third equation, but we can use it as a check, plug the coordinates into the third equation:
−7=−3(2)−1
−7=−6−1
−7=−7 it works.
then 
then 
, again by using Pythagoras theorem 
then 
then 
