The segment connecting the two midpoints of the sides of a triangle is called its midsegment or midline. The midsegment of a triangle has interesting characteristics:
(1) the midsegment connecting the midpoints of the two sides of a triangle is parallel to the third side and (2) its length is also half of the third side. In this post, we prove these two theorems.
In the given above, is a triangle and is a midsegment.
The segment connecting the two sides of a triangle is parallel to and half the length of the third side.
What To Show
- is parallel to
Let be a triangle with points , , and having coordinates , and respectively as shown in the figure above.
Since is the midpoint of , by the midpoint formula,
Now, is also the midpoint of , so
It follows that the slope of
and the slope of
Since and have the same slope, they are parallel. We have proved the first theorem.
Now, it remains to show that is half that of .
By the distance formula,
For the length of , we have