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?

Median of a Triangle

The median of a triangle is a line segment that connects one corner (vertex) of the triangle to the middle point of the opposite side. Each triangle has three medians, one from each vertex. These medians always meet at a special point inside the triangle called the centroid. The centroid acts like the triangle’s balance point.

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

Median of a Triangle Formula

There is a formula to find the length of a median if you know the lengths of all three sides of a triangle. The formula is:  Ma=2b2+2c2−a24. Where  a is the side opposite to the median, and b and c are the other two sides. This formula helps calculate how long a median is without measuring it directly.

Length of Median Formula

To find the length of a median, you use the formula based on the triangle’s side lengths. For example, the median from vertex A to the opposite side BC can be found by plugging the side lengths into:  Ma=2b2+2c2−a24. This formula comes from a useful rule in geometry called Apollonius’s theorem.

Enroll Now for Kids Online Classes

Median of a Triangle Properties

• 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

Read More -  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