What is the orthocenter of a triangle with corners at #(9 ,7 )#, #(4 ,4 )#, and (8 ,6 )#?
2 Answers
See below.
Explanation:
We will call the vertices
We need to find two equations that are perpendicular to two sides and pass through two of the vertices. We can find the slope of two of the sides and consequently the slope of the two of the perpendicular lines.
Slope of AB:
Slope perpendicular to this:
This has to pass through vertex C, so equation of line is:
Slope of BC:
Slope perpendicular to this:
This has to pass through vertex A, so equation of line is:
Where [1 ] and [ 2 ] intersect is the orthocenter.
Solving [1] and [2] simultaneously:
Using [2]:
Orthocenter:
Because the triangle is obtuse the orthocenter is outside of the triangle. this can be seen if you extend the altitude lines until they cross.
Orthocenter
Circumcenter
Explanation:
Orthocenter
Given
Those vectors are easily obtained, For example
Now we have
Those three lines intersect at the triangle's orthocenter
Choosing
giving the equations
Now solving for
and then
Circumcenter
The circumference equation is given by
now if
subtracting the first from the second
subtracting the first from the third
giving the system of equations
Now substituting the given values we get at
Attached a plot showing the orthocenter (red) and the circumcentercenter (blue).