Question

Recall that the median AM to side BC in triangle ABC is the line segment with endpoints at A and M that bisects side BC. Show that the median AM has a length that is smaller than the average of the lengths of the sides AB and AC

Answer 1

The median AM has a length smaller than the average of the lengths of sides AB and AC.

Step-by-step explanation:

To show that the median AM has a length smaller than the average of the lengths of sides AB and AC, we can use the triangle inequality theorem. According to this theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

In triangle ABC, let AB be the longest side. Since M is the midpoint of side BC, the median AM must be shorter than AB.

Now, let's consider the average of the lengths of sides AB and AC. This average would be (AB + AC)/2. Since AB is the longest side, (AB + AC)/2 will be greater than AB.

Therefore, we can conclude that the median AM has a length that is smaller than the average of the lengths of sides AB and AC.