5.4 Medians and Altitudes

Home , Altitude (triangle), Median (geometry)

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.)