1. GEOMETRY Medians and altitudes of a Triangle

2. Median of a triangle
A median of a triangle is a segment from a vertex to the midpoint of the opposite side.
median
median
median

3. Centroid of a triangle B C D E F P
The medians of a triangle intersect at the centroid, a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. The centroid of a triangle can be used as its balancing point. A
This is also a 2:1 ratio

4. Solving problems from involving medians and centroids
If you know the length of the median
P is the centroid, find BP and PE, given BE = 48 A B C D E F P
2 : 1
so BP=32
PE=16

5. Solving problems from involving medians and centroids
If you know the length of the segment from the vertex to the centroid
P is the centroid, find AD and PD given AP = 6 A B C D E F P

6. Solving problems from involving medians and centroids
If you know the length of the segment from the midpoint to the centroid
P is the centroid, find AD and AP given PD = 9 A B C D E F P

7. Find the Centroid on a Coordinate Plane
SCULPTURE An artist is designing a sculpture that balances a triangle on top of a pole. In the artist’s design on the coordinate plane, the vertices are located at (1, 4), (3, 0), and (3, 8). What are the coordinates of the point where the artist should place the pole under the triangle so that it will balance?
You need to find the centroid of the triangle. This is the point at which the triangle will balance.

8. Find the Centroid on a Coordinate Plane

9. Your Turn
BASEBALL A fan of a local baseball team is designing a triangular sign for the upcoming game. In his design on the coordinate plane, the vertices are located at (–3, 2), (–1, –2), and (–1, 6). What are the coordinates of the point where the fan should place the pole under the triangle so that it will balance?

10. altitude of a triangle
An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
Every triangle has three altitudes. An altitude can be inside, outside, or on the triangle.

11. Altitude of an Acute Triangle
Point of concurrency “P” or orthocenter
The point of concurrency called the orthocenter lies inside the triangle.

12. Altitude of a Right Triangle
The two legs are the altitudes
The point of concurrency called the orthocenter lies on the triangle.
Point of concurrency “P” or orthocenter

13. Altitude of an Obtuse Triangle
The point of concurrency of the three altitudes is called the orthocenter
The point of concurrency lies outside the triangle.

14. orthocenter of the triangle
In ΔQRS, altitude QY is inside the triangle, but RX and SZ are not. Notice that the lines containing the altitudes are concurrent at P. This point of concurrency is the orthocenter of the triangle.

15. Special Segments in Triangles
Name
Type
Point of Concurrency
Center Special Quality
From / To
Median
segment
Centroid
Center of Gravity
Vertex midpoint of segment
Altitude
Orthocenter
none
Vertex none

16. try it ALGEBRA Points U, V, and W are the midpoints
of respectively. Find a, b, and c.