What is the orthocenter of a triangle with corners at #(4 ,3 )#, #(9 ,5 )#, and (8 ,6 )#?

Question

What is the orthocenter of a triangle with corners at #(4  ,3  )#, #(9 ,5   )#, and (8   ,6   )#?

Theodore Abad · Accepted Answer

Using the corners of the triangle, we can get the equation of each perpendicular; using which, we can find their meeting point #(54/7,47/7)#.

The rules we're gonna use are: The given triangle has corners A,B, and C in the order given above. The slope of a line that passes through #(x_1,y_1) , (x_2,y_2)# has slope = #(y_1-y_2)/(x_1-x_2)# Line A which is perpendicular to line B has #"slope"_A=-1/"slope"_B # The slope of: Line AB =#2/5# Line BC =#-1# Line AC =#3/4# The slope of the line perpendicular to each side: Line AB =#-5/2# Line BC =#1# Line AC =#-4/3# Now you can find the equation of each perpendicular bisector passing through the opposite corner. For example, the line perpendicular to AB passing through C. They are, in the order used above: #y-6=-5/2(x-8)# #y-3=x-4# #y-5=-4/3(x-9)# If you solve any two of these 3, you'll get their meeting point-the orthocenter. Which is #(54/7,47/7)#.

Claire Akin · Answer

undefined The orthocenter of the triangle with corners at (4, 3), (9, 5), and (8, 6) is the point of intersection of the three altitudes of the triangle. To find the orthocenter, you need to determine the equations of the altitudes and then solve for their point of intersection.