9.02 Altitudes, medians, and bisectors of triangles
Exploration
The applet below contains a triangle with several special segments drawn. Move the vertex of the triangle around and click the checkboxes to show and hide each special segment type. Let's think about the answers to the following questions.
- How might we define each special type of segment?
- What properties does each special segment have?
- Which segments are always inside the triangle? Which ones are not?
- What has to happen for the special segments to coincide?
Triangle altitudes
No matter where we move the points of the triangle, the altitude is always a segment that is perpendicular to the base and passes through the vertex.
An altitude of a triangle is a segment from any vertex and perpendicular to the line containing the opposite side.
By its definition, the altitude does not need to be inside the triangle. It might even coincide with one of the sides of the triangle.
An altitude may be inside, outside, or one of the sides of a triangle.
Triangle medians
No matter where we move the points in the applet, the median is always inside the triangle. The median is the segment that connects one of the vertices of the triangle to the midpoint of the opposite side.
By definition, a median will always be located inside a triangle.
A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side.
Angle bisectors of triangles
Recall that an angle bisector is a ray from the vertex of the angle that divides an angle into two congruent angles. This definition holds for the angle bisectors of a triangle. By this definition, an angle bisector will always pass through the vertex of a triangle.
By definition, an angle bisector will always be located inside a triangle.
Perpendicular bisectors of triangles
We can see in the applet above that the perpendicular bisector of a triangle always intersects the midpoint of a side. It's also always at a right angle to it. That's what makes it the perpendicular bisector. However, we can also see that it doesn't always intersect the vertex of the opposite side.
A perpendicular bisector does not necessarily pass through the vertex.
Coinciding special segments
If we move the vertex of the triangle in the applet so that the triangle is isosceles, a special situation occurs. All the special segments overlap!
We can prove that this is true for any isosceles triangle using what we know about congruent triangles and their corresponding parts.
Practice questions
Question 1
In the triangle shown below, what does the dashed line represent?
An angle bisector
AA perpendicular bisector
BAn altitude
CA median
D
Question 2
In the triangle shown below, what does the dashed line represent?
An angle bisector.
AA perpendicular bisector.
BAn altitude.
CA median.
D
Question 3
Given $\overline{MQ}$MQ is an altitude to isosceles triangle $MRP$MRP, prove that $\overline{MQ}$MQ is also an angle bisector.
