Any triangle may be rotated and translated so that one vertex is at the origin and another vertex is on the positive x-axis. what is the relationship between the length of one of the triangle's midlines, and the

Since line segment DE is a midline of the triangle joining the midpoints of sides AB and AC, thus by the triangle midsegment theorem, line segment DE is parallel to side BC and is half the length of line segment BC.

The triangle mid-segment theorem states that the mid-segment of a triangle, which joins the midpoints of two sides of a triangle, is parallel to the third side of the triangle and half the length of that third side of the triangle.

Therefore, the relationship between the length of DE and the length of BC is that the length of DE is half the length of BC.