circumcenter, orthocenter and centroid?

Question

So, if you have an obtuse triangle with the coordinates being E(-8.00, 8.00) C(8.00,8.00) and D(0.00, -4.00) (E and D are base angles.) what are the equations to find the circumcenter, centroid and orthocenter? and where do the lines intersect at?

Answer 1

Using basic techniques, you should be able to find the coordinates of the midpoints of the sides. You should also be able to find the slope of each side. You'll need these pieces of data

The circumcenter is the point of intersection of the perpendicular bisectors. The perpendicular bisector of each side passes through its midpoint and is perpendicular to the side. You can find the coordinates of the midpoint easily, and the perpendicular line will have slope equal to the negative reciprocal of the slope of the side. You can get an equation for the line by using the point-slope form. Find equations for all three perpedicular bisectors, and then solve the system of equations to get their intersection.

The centroid is at the intersection of the medians. Each median joins a vertex of the triangle with the midpoint of the opposite side. You can get the equations for these lines using the point-point form, since you know the vertices and midpoints. Once again, solve the system of three equations.

Finally, the orthocenter is the point of intersection of the altitudes, which is a line that passes through a vertex and is perpendicular to the opposite side. Use the point-slope form to get the equations of all three lines and solve the system.

Answer 2

Circumcenter- the place the perpendicular bisectors meet, that's equidistant to the vertices Incenter- the place the perspective bisectors meet, that's equidistant to the perimeters Centroid- the place the medians meet, that's the balancing factor of a triangle, it spilts the area into 2 congruent areas Orthocenter- the place the altitudes meet