Midsegments of Triangles

Unit 5: Relationships Within Triangles · ~15–30 min

What is a midsegment?

A midsegment of a triangle is a segment that connects the midpoints of two sides. Every triangle has three midsegments — one for each pair of sides — and each one has a fixed, predictable relationship to the third side of the triangle.

• Triangle Midsegment Theorem: A midsegment is parallel to the third side of the triangle, and its length is exactly half the length of that third side.

That single theorem does two jobs at once: it guarantees a parallel relationship and a length relationship, which is why it shows up constantly in later proofs about parallel lines and proportional sides.

Guided practice

• 1.In triangle PQR, M and N are midpoints of PQ and PR. If QR = 24, find MN.
  Show answer

  MN = ½(24) = 12.
• 2.A midsegment measures 7.5. What is the length of the side it's parallel to?
  Show answer

  The parallel side is twice the midsegment: 2(7.5) = 15.
• 3.Midsegment DE = 4x + 1, and the third side is 34. Find x.
  Show answer

  DE must equal ½(34) = 17, so 4x + 1 = 17, 4x = 16, x = 4.
• 4.Explain why a midsegment can never be longer than the side it's parallel to.
  Show answer

  The Midsegment Theorem fixes the midsegment's length at exactly half the third side, so it is always shorter — it can never equal or exceed the full side.
• 5.Triangle XYZ has all three midsegments drawn, forming a smaller triangle inside. If XY = 10, YZ = 14, and XZ = 18, find the perimeter of the midsegment triangle.
  Show answer

  Each midsegment is half of one original side: 5 + 7 + 9 = 21.