The midpoint theorem states that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length. True or false?

• A. False
• B. True
• C. Only for isosceles triangles
• D. Only if the triangle is right-angled

Answer: (B)

The segment joining the midpoints of two sides of a triangle is parallel to the third side and has half its length. This holds for any triangle, not just special ones. If you mark the midpoints on two sides, say D and E on AB and AC, the smaller triangle ADE is similar to the whole triangle ABC with a scale factor of 1/2. That means DE corresponds to BC and is exactly half as long, and DE runs in the same direction as BC, so DE is parallel to BC. Since this works for any triangle, the statement is true.