1. 5-4 Medians and Altitudes

2. Median of a Triangle
A segment whose endpoints are a vertex and the midpoint of the opposite side.

3. A triangle’s three medians are always concurrent, and are always concurrent on the interior of the triangle.
In a triangle, the point of concurrency of the medians is the centroid of the triangle. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance.

8. Centroid to vertex is always a longer segment than centroid to midpoint.
COMPARISONS:
Portion of median to portion of median
½ or 2/1 ratios
Whole Median to portion of median
1/3 or 2/3 ratios (reciprocals are possibility as well)
When setting up equations deal only with the segments that you have information for within the original problem. We want to make the parts provided equal to one another, obviously one part needs to be adjusted through multiplication in order to make them equal. How we decide to adjust is decided by the above information.

9. Altitude of a triangle
The perpendicular segment from a vertex of the triangle to the line containing the opposite side.
An altitude of a triangle can be inside the triangle or outside, or in right triangles it is a side of the triangle.

11. The lines that contain the altitudes of a triangle are concurrent at the orthocenter of the triangle. The orthocenter of a triangle can be inside, on(for right triangles the orthocenter will be the vertex at the right angle), or outside the triangle.

12. Orthocenter with coordinate geometry
Need to use a system of equations
The equation of an altitude line needs to use a vertex as (x1,y1) and then the opposite signed reciprocal of the slope of the line that the altitude is going towards.
You need two altitude equations
Set them equal to each other to find orthocenter.

13. Circumcenter with coordinate geometry
Need to use a system of equations
The equation of a perp. Bisector line needs to use a midpoint as (x1,y1) and then the opposite signed reciprocal of the slope of the line that the perp. Bisector is intersecting
You need two perp. bisector equations
Set them equal to each other to find circumcenter.

14. Centroid with coordinate geometry
These can be found using systems of equations but it is easier to just add the x’s and divide by 3, then add the y’s and divide by 3.
Average of x’s and y’s