Lesson 6.3 Medians and Altitudes of Triangles

Median of a Triangle

1. Segment Bisector vs. Perpendicular Bisector

   • A segment bisector intersects a segment at its midpoint and can run in any direction.

   • A perpendicular bisector is a segment bisector that is also perpendicular to the segment.

   • To construct a segment bisector, follow the steps for a perpendicular bisector, but instead of drawing the perpendicular line, just mark the midpoint.

2. Definition of a Median

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

   • Every triangle has three medians.

3. Centroid

   • All three medians of a triangle intersect at one point called the centroid.

   • The centroid is the concurrent point of the medians.

4. Centroid Theorem

   • The centroid divides each median into two segments, one of which is two-thirds of the entire length of that same median. The longer part (from the vertex to the centroid) is $2/3$ of the total median length, and the shorter part (from the centroid to the midpoint) is $1/3$ of the total median length.

   • For instance, if segment AD is a median with centroid Q, then $AQ = (2/3)AD$ and $QD = (1/3)AD$ or $AQ = 2QD$.

5. Coordinate Plane Tips

   • Look for vertical or horizontal medians to avoid the distance formula if possible.

   • The midpoint formula will frequently be used due to working with midpoints of sides.

Altitude of a Triangle

1. Definition of an Altitude

   • The altitude (or height, $h$) of a triangle is a segment from a vertex perpendicular to the opposite side, or to the line that contains the opposite side.

   • In an obtuse triangle, an altitude may be exterior to the triangle by extending the opposite side.

   • Every triangle has three altitudes.

2. Orthocenter

   • The three altitudes of a triangle intersect at a point of concurrency called the orthocenter.

3. Location of the Orthocenter

   • Acute triangle: Orthocenter is inside the triangle.

   • Obtuse triangle: Orthocenter is outside the triangle.

   • Right triangle: Orthocenter is on the triangle (at the vertex of the right angle).

4. Area Formula

   • The area of a triangle is given by $A = (1/2)bh$, where $b$ is the base and $h$ is the altitude (height) perpendicular to that base ($b \perp h$).

Special Case: Isosceles Triangle

• In an isosceles triangle, the perpendicular bisector, angle bisector, median, and altitude from the vertex angle to the base are all the same segment.