Definition: A midsegment of a triangle joins the midpoints of two sides; it is parallel to the third side and exactly half as long.
What is the midsegment theorem?
Take triangle ABC with D the midpoint of AB and E the midpoint of AC. Then DE is parallel to BC, and DE = (1/2)BC. A triangle has three midsegments, which together cut it into four congruent triangles.
How is it used?
To find unknown lengths, set the midsegment equal to half the third side. To prove parallel lines, show that both endpoints are midpoints.
Example. In triangle PQR, M and N are the midpoints of PQ and PR, with MN = 3x + 1 and QR = 10x – 14. Then 3x + 1 = (1/2)(10x – 14) = 5x – 7, so 8 = 2x and x = 4. So MN = 13 and QR = 26. The perimeter of the medial triangle formed by all three midsegments is half the perimeter of the original.
What errors occur?
Doubling when you should halve. Using a segment that is parallel to a side but not joining two midpoints also fails. In coordinates, check the slopes and the lengths to confirm. A trapezoid has its own midsegment, with a different formula.