1. GEOMETRYMedians 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 C • 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. E D P B A F 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 C E D 2 : 1 so BP=32 PE=16 P B A F

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 C E D P B A F

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 C E D P B A F

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 altitude altitude The point of concurrency of the three altitudes is called the orthocenter The point of concurrency lies outside the triangle.

14. 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. orthocenter of the triangle

15. Special Segments in Triangles

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