Altitudes Medians and Angle Bisectors

Just as there are special names for special types of triangles, so there are special names for special line segments within triangles. Now isn't that kind of special?

Base and altitude

Every triangle has three bases (any of its sides) and three altitudes (heights). Every altitude is the perpendicular segment from a vertex to its opposite side (or the extension of the opposite side) (Figure 1).




Figure 1 Three bases and three altitudes for the same triangle.



Altitudes can sometimes coincide with a side of the triangle or can sometimes meet an extended base outside the triangle. In Figure 2, AC is an altitude to base BC , and BC is an altitude to base AC .


Figure 2 In a right triangle, each leg can serve as an altitude.


In Figure 3, AM is the altitude to base BC .





Figure 3 An altitude for an obtuse triangle.


It is interesting to note that in any triangle, the three lines containing the altitudes meet in one point (Figure 4).





Figure 4 The three lines containing the altitudes intersect in a single point, 

which may or may not be inside the triangle.


Median

median in a triangle is the line segment drawn from a vertex to the midpoint of its opposite side. Every triangle has three medians. In Figure 5, E is the midpoint of BC . Therefore, BE = EC. AE is a median of Δ ABC.




Figure 5 
A median of a triangle.

In every triangle, the three medians meet in one point inside the triangle (Figure 6).





Figure 6 
The three medians meet in a single point inside the triangle.

Angle bisector

An angle bisector in a triangle is a segment drawn from a vertex that bisects (cuts in half) that vertex angle. Every triangle has three angle bisectors. In Figure , is an angle bisector in Δ ABC.




Figure 7 
An angle bisector.


In every triangle, the three angle bisectors meet in one point inside the triangle (Figure 8).




Figure 8 
The three angle bisectors meet in a single point inside the triangle.


In general, altitudes, medians, and angle bisectors are different segments. In certain triangles, though, they can be the same segments. In Figure , the altitude drawn from the vertex angle of an isosceles triangle can be proven to be a median as well as an angle bisector.





Figure 9 
The altitude drawn from the vertex angle of an isosceles triangle.

Example 1: Based on the markings in Figure 10, name an altitude of Δ QRS, name a median of Δ QRS, and name an angle bisector of Δ QRS.





  Figure 10 
Finding an altitude, a median, and an angle bisector.



RT is an altitude to base QS because RTQS .



SP is a median to base QR because P is the midpoint of QR .

QU is an angle bisector of Δ QRS because it bisects ∠ RQS.