What is the orthocenter of a triangle with corners at #(2 ,6 )#, #(9 ,1 )#, and (5 ,3 )#?

1 Answer
Mar 31, 2017

The Orthocenter is #(-10,-18)#

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 #(2,6)# to #(9,1) # is:

#m_1 = (1-6)/(9-2)#

#m_1 = -5/7#

The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:

#p_1 = -1/m_1#

#p_1 = -1/(-5/7)#

#p_1 = 7/5#

The altitude must pass through point #(5,3)#

We can use the point-slope form for the equation of a line to write the equation for the altitude:

#y = 7/5(x-5)+3#

Simplify a bit:

#y = 7/5x-4" [1]"#

The slope of the line segment from point #(2,6)# to #(5,3) # is:

#m_2 = (3-6)/(5-2)#

#m_2 = -3/3#

#m_2 = -1#

The slope of the altitude drawn through this line segment will be perpendicular, which means that the perpendicular slope will be:

#p_2 = -1/m_2#

#p_2 = -1/(-1)#

#p_2 = 1#

The altitude must pass through point #(9,1)#

We can use the point-slope form for the equation of a line to write the equation for the altitude:

#y = 1(x-9)+1#

Simplify a bit:

#y = x-8" [2]"#

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]:

#7/5x-4 = x-8#

Solve for the x coordinate of intersection:

#2/5x =-4#

#x = -10#

To find the value of y, substitute -10 for x into equation [2]:

#y = -10 - 8#

#y = -18#

The Orthocenter is #(-10,-18)#