Midsegment of a Triangle
In short: The midsegment of a triangle joins the midpoints of two sides, runs parallel to the third side, and measures exactly half as much as that side.
GeometryGeometryG-CO.C.10
What is Midsegment of a Triangle?
The midsegment of a triangle is a segment drawn between the midpoints of two of its sides. A triangle has three of them, one for each pair of sides, and together they cut it into four small congruent triangles.
The midsegment theorem has two parts. The segment is parallel to the third side, and its length is half of the third side: if \(D\) and \(E\) are midpoints of \(\overline{AC}\) and \(\overline{BC}\), then \(\overline{DE} \parallel \overline{AB}\) and \(DE = \frac{1}{2}AB\).
How to spot it
- A figure marks the middle of two sides with tick marks and joins those marks by a segment.
- The words midsegment, midline or joins the midpoints appear in the statement.
- One length is given by an expression in x, and the third side by another expression, with a half relationship hidden between them.
- A problem asks for a length across a triangular sail, lot or frame without measuring it directly.
The method
- Identify the two midpoints and the side of the triangle that does not touch the segment.
- Write the relation: the joining segment equals half of the opposite side, or the side is double it.
- Substitute any expressions, solve for the unknown, and compute the lengths.
- Use the parallel property if angles are needed: the joining segment makes equal corresponding angles with the base.
Midsegment of a Triangle: solved example
Problem
On a triangular sail, a seam \(\overline{DE}\) joins the midpoints of two sides and runs parallel to the bottom edge. The seam measures \((2x + 11)\) inches and the bottom edge measures \((6x + 4)\) inches. Find both lengths.
Solution
The seam is half the bottom edge, so the bottom edge is double the seam: \(6x + 4 = 2(2x + 11)\). Expand: \(6x + 4 = 4x + 22\), so \(2x = 18\) and \(x = 9\).
Seam: \(2(9) + 11 = 29\) inches. Bottom edge: \(6(9) + 4 = 58\) inches. Check: \(58 \div 2 = 29\). The lengths are 29 inches and 58 inches.
Try it
Question 1
A midsegment of a triangle measures 17.5 inches. How long is the side it is parallel to?
Show the answer
That side is double the midsegment: \(2 \times 17.5 = 35\) inches.
Question 2
A triangle has vertices \(P(0, 0)\), \(Q(12, 0)\) and \(R(4, 10)\). Find the length of the midsegment joining the midpoints of \(\overline{PR}\) and \(\overline{QR}\).
Show the answer
The midpoints are \((2, 5)\) and \((8, 5)\), which are 6 apart. This is half of \(PQ = 12\), as expected.
Midsegment of a Triangle vs. Centroid
A midsegment joins two midpoints and lies parallel to a side. A median connects a vertex to the midpoint of the opposite side, and the three medians meet at the centroid. Both use midpoints, but one segment sits inside the triangle’s middle band, while the other slices from a corner.
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