Median of a Triangle: Definition, Formula, and Examples

Median of a Triangle: What happens when you draw a line from a vertex of a triangle to the midpoint of the opposite side?

What is the Median of a Triangle?

• D is the midpoint of side BC
• The line segment AD is the median because it connects vertex A to midpoint D, dividing side BC into two equal parts: BD=DC

Median of Triangle Definition

International Numeral System

Properties of the Median of a Triangle

• Each median connects a vertex to the midpoint of the opposite side, dividing that side into two equal parts.
• Every triangle has three medians, one from each vertex, ensuring symmetry across the shape.
• All three medians meet at a single point called the centroid. The centroid is the triangle's center of mass or balance point.
• The centroid divides each median into two parts in a 2:1 ratio. For example, if a median is 9 units long, the segment from the vertex to the centroid is 6 units, and from the centroid to the midpoint is 3 units.

How to Find the Median of a Triangle?

Steps to Find the Median of a Triangle with Coordinates

Step 1: Identify the Coordinates : Begin by noting the coordinates of the triangle's vertices. Denote them as (x ₁ ,y ₁ ), (x ₂ ,y ₂ ), and (x ₃ ,y ₃ )

Step 2: Calculate the midpoint of the opposite side. Choose a vertex for which you want to calculate the median, and identify the opposite side. Use the midpoint formula to find the midpoint of this side.

Midpoint = (x ₂ + x ₃ /2, y ₂ + y ₃ /2 )

d = √(x ₁ −Midpoint _x ) ² +  (y ₁ −Midpoint _y ) ²

Perfect Square

Steps to Construct a Median of a Triangle

Step 1: Draw the triangle and label its vertices A , B , and C .

Step 2: Find the midpoint of each side by measuring or using the midpoint formula.

Step 3: Connect each midpoint to the opposite vertex using a straight line.

Step 4: Verify that the three lines intersect at a single point, the centroid.

Median of Equilateral Triangle