Median of a Triangle

A median of a triangle is a line segment drawn from a vertex to the midpoint of the opposite side. Each triangle has three medians, one from each vertex to the midpoint of its opposite side. :contentReference[oaicite:0]{index=0}

      A
     / \
    /   \
   B---D---C
      (D is midpoint of BC)
   AD is a median

Definition: In triangle \(\triangle ABC\), if \(D\) is the midpoint of side \(BC\), then the line segment \(AD\) is called the median from vertex \(A\). Because all vertices have such a segment, there are three medians in total. :contentReference[oaicite:1]{index=1}

• Each median bisects the side it is drawn to (divides it into two equal parts). :contentReference[oaicite:2]{index=2}
• All three medians of a triangle meet at a single point called the centroid. :contentReference[oaicite:3]{index=3}
• The centroid divides each median in the ratio 2 : 1, counting from the vertex to the midpoint of the opposite side. :contentReference[oaicite:4]{index=4}
• The medians divide the triangle into six smaller triangles of equal area. :contentReference[oaicite:5]{index=5}

To construct a median from a vertex:

1. Identify the opposite side of the triangle.
2. Find the midpoint of that side (using a ruler or compass).
3. Draw a straight line from the vertex to that midpoint.

If \(D\) is the midpoint of \(BC\) in \(\triangle ABC\), then \(AD\) is the median.

The point where all three medians meet is called the centroid of the triangle. It has special properties:

• The centroid lies inside the triangle regardless of shape.
• It is the triangle’s “centre of mass” (if the triangle were a uniform sheet of material). :contentReference[oaicite:6]{index=6}
• Each median is divided by the centroid so that the portion from vertex to centroid is twice the length of the portion from centroid to midpoint of the side.

Medians are used in many geometry problems and real-life applications:

• Finding the centroid helps in balance and structural design.
• Using medians, one can prove relationships in triangles and compute areas.
• In coordinate geometry and physics, the centroid formula often uses medians.

Understanding medians strengthens your grasp of triangle geometry and helps with advanced topics.