Exercise 3 Page 361

Medians

From Exploration 1, we notice that the point of intersection of the three medians cut each median in a ratio of 2 to 1. Consider the following image focusing on median BF.

Conjectures Regarding Medians:

1. The three medians of a triangle will cross at some point D.
2. The point of concurrency D is 23 the length of the median from the vertex B.
3. The point of concurrency D is 13 the length of the median from the midpoint of the opposite side F.
4. This means the following equations are true.

BD=2/3BF [0.8em] DF=1/3BF [0.8em] DF=1/2BD

Altitudes

From Exploration 2, we notice that the altitude is always perpendicular to the opposite side. Also, the altitude can be both outside, inside or on the triangle depending the types of angles in the triangle.

Conjectures Regarding Altitudes:

1. The three altitudes of any triangle will cross.
2. For an acute triangle, this point of concurrency will be on the inside of the triangle.
3. For an right triangle, this point of concurrency will on the triangle.
4. For an obtuse triangle, this point of concurrency will be on the outside of the triangle.