The mid-segment of a triangle is the segment joining the midpoints of two sides. This segment is parallel to the third side and has what length relation to that side?

• A. parallel to the third side and half as long.
• B. perpendicular to the third side.
• C. lies on the third side.
• D. has the same length as the third side.

Answer: (A)

The mid-segment theorem tells us the segment joining the midpoints of two sides is parallel to the third side and has half its length. If M and N are the midpoints of AB and AC, then MN is parallel to BC. The triangles AMN and ABC are similar because they share angle A and MN is parallel to BC, making the other corresponding angles equal. Since AM = AB/2 and AN = AC/2, the similarity ratio is 1/2, so MN corresponds to BC with that same factor: MN = BC/2. Thus the mid-segment runs parallel to the third side and is half as long as that side.