Math Resources Hub

Midsegment

Mark the middle of two sides of a triangle and connect those two points. The short segment you draw — a — lines up with the third side in a way that never changes: it runs perfectly parallel to it, and it is exactly half its length.

ABCMNMN = ½ · BC
M and N are the midpoints of two sides. The midsegment MN is parallel to the base BC (the chevrons match) and half as long: MN = ½ · BC.

Explore

Reshape the triangle and watch the midsegment MN against the third side BC. Compare their lengths, and compare their directions. What two things stay true no matter how you drag?

Reshape the triangle with the sliders and watch the midsegment stay parallel to the base at exactly half its length. Then press “Show why”.

8.0
6.0
55°

The midsegment is 4.0, which is half of AB (8.0), and it is parallel to AB.

A triangle whose midsegment parallel to side AB joins the midpoints of the other two sides. It is always parallel to AB and exactly half as long: here the midsegment is 4.0 and AB is 8.0. Press “Show why” to watch a dilation by 2 from the opposite vertex carry the midsegment out onto AB.

Make a prediction

Make a prediction: In a triangle, the third side (the one the midsegment doesn't touch) is 12 cm long. How long is the midsegment joining the midpoints of the other two sides?

Test it

In the tool, slide the triangle through tall, flat, and lopsided shapes. The midsegment and the third side stay parallel — same tilt — the whole time, and the midsegment's length stays glued to exactly half. This pair of facts is the Triangle Midsegment Theorem.

Why it works

Because the midsegment is parallel to the third side, it cuts the top of the triangle into a smaller triangle with the same angles as the whole — the parallel line makes equal corresponding angles. That smaller triangle is a scaled-down copy of the big one. Each of its two sides runs from a vertex to a midpoint, so it is built at exactly half scale — which forces its bottom edge, the midsegment, to be half of the matching full side.

Key result

A midsegment joins the midpoints of two sides. It is parallel to the third side and half its length: midsegment=12 (third side)\text{midsegment} = \tfrac{1}{2}\,(\text{third side}).

Worked example: In a triangle, a midsegment measures 9 cm. Find the length of the side it is parallel to, and state how the two are oriented.

  1. The midsegment is half of the third side, so the third side is double the midsegment: 2×9=182 \times 9 = 18 cm.

    Before the next step: How long is the side the midsegment is parallel to?

Practice

Score: 0/4

  1. 1.The third side of a triangle is 16 cm. How long is the midsegment parallel to it, in cm?
  2. 2.A midsegment of a triangle is 9 cm long. How long is the side it is parallel to, in cm?
  3. 3.A midsegment measures (2x − 3) and the side it is parallel to measures 18. Find x.
  4. 4.A midsegment is parallel to the third side. Where the third side meets one of the cut sides, the angle is 50°. What is the corresponding angle where the midsegment meets that same side?
For teachers

Standard, common misconceptions, and suggested use will live here.