# Orthocenter and Centroids

• Jan 8th 2009, 05:02 PM
lethalxxlover
Orthocenter and Centroids
I need to find the orthocenter and centroid of a triangle with the coordinates

a(4,6) b(0,5) and c(-3,2)

help??
• Jan 9th 2009, 12:58 AM
Isomorphism
Quote:

Originally Posted by lethalxxlover
I need to find the orthocenter and centroid of a triangle with the coordinates

a(4,6) b(0,5) and c(-3,2)

help??

Here's the idea: Centroid is the meeting point of the medians and orthocenter is the meeting point of the altitudes..

Take it as an exercise to show that: if $\displaystyle A(x_1,y_1), \, B(x_2,y_2), \, C(x_3,y_3)$ are the vertices of a triangle, then the centroid is $\displaystyle \left(\frac{x_1+x_2+x_3}3,\frac{y_1+y_2+y_3}3\righ t)$.

As for the orthocenter. Find the equation of the line BC and then find the equation of the line that has a slope of $\displaystyle -\frac1{m_{BC}}$ (where $\displaystyle m_{BC}$ is the slope of BC) and passes through A.

Do the same thing for another vertex and get the equation of another altitude. Now get the meeting point of the intersection which is the required orthocenter...