2Median of a triangleA median of a triangle is a segment from a vertex to the midpoint of the opposite side.medianmedianmedian
3Centroid of a triangleBCDEFPThe 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.AThis is also a 2:1 ratio
4Solving problems from involving medians and centroids If you know the length of the medianP is the centroid, find BP and PE, given BE = 48ABCDEFP2 : 1so BP=32PE=16
5Solving problems from involving medians and centroids If you know the length of the segment from the vertex to the centroidP is the centroid, find AD and PD given AP = 6ABCDEFP
6Solving problems from involving medians and centroids If you know the length of the segment from the midpoint to the centroidP is the centroid, find AD and AP given PD = 9ABCDEFP
7Find 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.
9Your TurnBASEBALL 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?
10altitude of a triangleAn 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.
11Altitude of an Acute Triangle Point of concurrency “P” or orthocenterThe point of concurrency called the orthocenter lies inside the triangle.
12Altitude of a Right Triangle The two legs are the altitudesThe point of concurrency called the orthocenter lies on the triangle.Point of concurrency “P” or orthocenter
13Altitude of an Obtuse Triangle The point of concurrency of the three altitudes is called the orthocenterThe point of concurrency lies outside the triangle.
14orthocenter 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.
15Special Segments in Triangles NameTypePoint of ConcurrencyCenter Special QualityFrom / ToMediansegmentCentroidCenter of GravityVertex midpoint of segmentAltitudeOrthocenternoneVertex none
16try it ALGEBRA Points U, V, and W are the midpoints of respectively. Find a, b, and c.