Final answer:
To find the endpoints of the midsegment of triangle LMN that is parallel to side LN, calculate the midpoints of sides LM and MN using the midpoint formula, then connect these midpoints to determine the midsegment.
Step-by-step explanation:
The Triangle Midsegment Theorem states that the midsegment of a triangle is parallel to one of its sides and is half as long as that side. To find the coordinates of the endpoints of the midsegment that is parallel to LN in △LMN, you must first determine the midpoint coordinates of LM and MN. Let's denote the midpoint of LM as M1 and the midpoint of MN as M2. If L(x1, y1), M(x2, y2), and N(x3, y3) are the coordinates of the vertices, then the midpoint formula, which is ((x1 + x2)/2, (y1 + y2)/2), can be applied to find M1 and M2. The line segment joining M1 and M2 would be the required midsegment.
The coordinates for M1 would be ((x1 + x2)/2, (y1 + y2)/2) and for M2 ((x2 + x3)/2, (y2 + y3)/2). Therefore, the endpoints of the midsegment parallel to LN are M1 and M2.