Median of a Triangle
Understand medians, the centroid, and their properties in triangles
Overview
A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. Medians have special geometric properties and play an important role in understanding triangle structure.
A median of a triangle is a straight line segment joining a vertex to the midpoint of the opposite side. Unlike heights, medians do not form right angles with the sides they intersect.
Every triangle has exactly three medians, one from each vertex. All three medians intersect at a single point called the centroid (center of gravity).
Properties of Medians
Medians in triangles have several important geometric properties that distinguish them from other triangle line segments.
Key Properties
- Every triangle has exactly three medians
- All three medians intersect at a single point called the centroid
- The centroid is always inside the triangle
- Each median divides the triangle into two smaller triangles of equal area
- The centroid divides each median in the ratio 2:1 from vertex to opposite side
- Unlike heights, medians do not form right angles with the opposite side
- The centroid is the center of mass of the triangle (for uniform density)
The Centroid (Center of Gravity)
The centroid, also called the center of gravity, is the point where all three medians intersect. It is one of the most important special points in a triangle.
Centroid Properties
The centroid divides each median into two segments. The distance from the vertex to the centroid is twice the distance from the centroid to the opposite side's midpoint.
If \(P\) is the centroid and the median connects vertex \(A\) to midpoint \(M\):
In the diagram above: Point \(P\) is the centroid. It divides each of the three medians in the ratio 2:1, with the longer segment always being from the vertex.
Median Length Formulas
The lengths of the three medians in a triangle can be calculated using formulas based on the side lengths. For a triangle with sides \(a\), \(b\), and \(c\):
Median to Side a
Formula for \(m_a\)
Median to Side b
Formula for \(m_b\)
Median to Side c
Formula for \(m_c\)
- \(m_a\) = median from vertex \(A\) to the midpoint of side \(a\)
- \(m_b\) = median from vertex \(B\) to the midpoint of side \(b\)
- \(m_c\) = median from vertex \(C\) to the midpoint of side \(c\)
- \(a\), \(b\), \(c\) = lengths of the three sides
Area Division Property
One of the most useful properties of medians is that they divide a triangle into regions of equal area. This property is fundamental in many geometric applications.
- Each median divides the triangle into two triangles of equal area
- All three medians together divide the triangle into six smaller triangles of equal area
- Each smaller triangle has area = \(\frac{1}{6}\) of the original triangle
Comparison: Medians vs Other Triangle Elements
Median
Connects: Vertex to midpoint of opposite side
Angle: Not necessarily perpendicular
Height
Connects: Vertex perpendicular to opposite side
Angle: Always 90°
Angle Bisector
Divides: Vertex angle into equal parts
Angle: Bisects vertex angle
Key Points
- A median connects a vertex to the midpoint of the opposite side
- Every triangle has exactly three medians
- All three medians intersect at the centroid (center of gravity)
- The centroid is always inside the triangle
- The centroid divides each median in the ratio 2:1
- Each median divides the triangle into two equal areas
- Three medians divide the triangle into six equal smaller triangles
- Medians do not form right angles (unlike heights)
- The centroid is the center of mass for uniform density
|
|