5.4 Medians and Altitudes Segment connecting a vertex to the Median of a triangle: midpoint of the opposite side of a triangle. Centroid: Point of concurrency of the medians of a triangle. Master Card (Medians meet at the centroid.) (TH) The centroid is two thirds of the distance from each vertex to the midpoint of the opposite side. **That means that each median is divided into a small piece that is one third of the whole median and a longer piece that is twice as long as the short piece. Hint: 6 is the longer of the two pieces because it goes to the vertex. The shortest piece, ML, is half that; so ML = 3. GL is the total, or 3ML; so it is 9. Altitude: The perpendicular segment connecting a vertex to the line containing the opposite side of the triangle. The altitude can be inside the triangle. (all three altitudes in acute triangles.) pics drawn on board. The altitude can be a side of the triangle. (two of the altitudes in a right triangle.) The altitude can be outside the triangle. (two of the altitudes in an obtuse triangle.) Orthocenter: The point of concurrency of the altitudes. Atlantic Ocean Current: Altitudes meet at the Orthocenter. The theorem for altitudes only says that they are concurrent. Hint: Find the median of one side and use the Centroid theorem. The centroid is two thirds from point K to point M. P(4,4) Hint: draw a picture and state the given and prove. Since BD is an altitude, we have two right triangles with a reflexive leg. They are congruent by HL. By CPCTC, we know that AD = CD. By definition, BD is the median. Geometry Page 322 (1-12 left column, 25-27, 46-49, 53-55) Page 322( 14-22 even, 28, 33-35, 38, 40-42, 50-52) PBCC ABI MC AOC (draw a pic of each and state the rule......) Test (Info will also be on the first semester exam.) .