What is the orthocenter of a triangle with corners at #(2 ,6 )#, #(9 ,1 )#, and (5 ,3 )#?
1 Answer
The Orthocenter is
Explanation:
The Orthocenter of a triangle is the point of intersection of the 3 altitudes of the triangle.
The slope of the line segment from point
The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:
The altitude must pass through point
We can use the point-slope form for the equation of a line to write the equation for the altitude:
Simplify a bit:
The slope of the line segment from point
The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:
The altitude must pass through point
We can use the point-slope form for the equation of a line to write the equation for the altitude:
Simplify a bit:
We could repeat this process for the third altitude but we have already enough information to determine the intersection point.
Set the right side of equation [1] equal to the right side of equation [2]:
Solve for the x coordinate of intersection:
To find the value of y, substitute -10 for x into equation [2]:
The Orthocenter is