Median of a Triangle
In geometry, a median of a triangle refers to a line segment joining a vertex of the triangle to the midpoint of the opposite side, thus bisecting that side. For any triangle, there are exactly three medians, one from each vertex. These intersect each other at the triangle's centroid.
What is Median of a Triangle?
A line segment, joining a vertex to the midpoint of the side, opposite to that vertex, is called the median of a triangle. In the figure given below, AD is the median, dividing BC into two halves, such that, DB = DC.
Median of a Triangle Theorem
The median of a triangle theorem states that the medians of a triangle intersect at a point called the centroid of the triangle, which is twothirds of the distance from the vertices to the midpoint of the opposite sides. The centroid of a triangle is the point of intersection of all three medians of that triangle, with the fact that a triangle has exactly three medians.
Properties of Medians of a Triangle
 The median of a triangle further divides the triangle into two triangles having the same area measurement.
 For a given triangle, the second median divides the triangle formed by the first median in the ratio 1:2.
 Every triangle has 3 medians, one from each vertex. The point of concurrency of 3 medians forms the centroid of the triangle.
 Irrespective of the shape or size of a triangle, its three medians meet at a single point.
 Each median of a triangle divides the triangle into two smaller triangles that have equal areas. In fact, the 3 medians divide the triangle into 6 smaller triangles of equal area.
 The sum of two sides of a triangle is greater than the median drawn from the vertex, which is common.
Altitude and Median of Triangle
A median of a triangle is defined as a line segment that joins the vertex and the midpoint of the opposite side of the triangle. All triangles have 3 medians (one from each vertex), meeting at a single point, independent of the type of the triangle. The 3 medians meet at a point, a common point called the centroid of the triangle.
An altitude of a triangle is defined as a line segment joining the vertex to the opposite side of the triangle at a right angle. All triangles have 3 altitudes (one from each vertex), meeting at a single point, of the triangle is. The 3 altitudes meet at a point, that may lie inside or outside the triangle, known as the orthocenter of the triangle.
How to Find the Median of a Triangle?
Let's have a look on how to calculate the length of each median:
\(m_{a}=\sqrt{\frac{2 b^{2}+2 c^{2}a^{2}}{4}}\)
\(m_{b}=\sqrt{\frac{2 a^{2}+2 c^{2}b^{2}}{4}}\)
\(m_{c}=\sqrt{\frac{2 a^{2}+2 b^{2}c^{2}}{4}}\)
a, b and c = three sides of the triangle.
Median of a Triangle Equation
For a triangle whose coordinates of the three vertices are given, we can follow the steps given below to find the medians of the given triangle.
 Step 1: Find the coordinates of the vertices of the triangle.
 Step 2: Find the coordinates of midpoints of line segments.
 Step 3: Join the vertex to the midpoint of the opposite side of the triangle. You will get medians.
 Step 4: Find the equation of the median of a triangle using the formula used to find the equation of the straight line (twopoint form). Here the equation of the median AD of triangle ABC= \(\left(\mathbf{y}\mathbf{y}_{1}\right)=\left(\frac{\mathrm{y}_{2}\mathrm{y}_{1}}{\mathrm{x}_{2}\mathrm{x}_{1}}\right)_{\mathrm{AD}}\left(\mathrm{x}\mathrm{x}_{1}\right)\)
Therefore, the equations of the medians of the triangle ABC will be: (y−y1)=(y2−y1/x2−x1)(x−x1)
 AD: 7x + 12 = 4y
 Similarly, BF: x + 2y = 12
 And CE: x + 36 = 7y
Important Notes
 Each median divides the triangle into two smaller triangles that have the same area
 The centroid (the point where they meet) is the center of gravity of the triangle
 Sum of medians of a triangle: The sum of the squares of the medians of a triangle equals threefourths the sum of the squares of the sides of the triangle.
 The perimeter of a triangle is greater than the sum of its three medians.
 The corresponding sides, perimeters, medians, and altitudes, all will be in the same ratio, for two similar triangles.
 The midpoint of a segment divides it into two congruent segments, whereas the median of a triangle divides it into two congruent triangles.
 Also, if the two triangles are congruent. The medians of congruent triangles are equal as corresponding parts of congruent triangles are congruent.
Thinking Out Of the Box!
A sculpture is planning to make a new sculpture, that consists of a triangle balanced on another triangle. Which point could be the point of balance? Does finding centroid would help?
Articles Related to Median of a Triangle
Solved Examples on Median of a Triangle

Example 1: Give a term that describes the point O, shown in the figure given below.
Solution:
Point O is the center of the triangle and the point of intersection of all the three medians of a triangle. Therefore, the given point O is the Centroid of the triangle.

Example 2: For the given triangle ABC, G is the centroid and BC = 10 units. Determine the length of BD.
Solution:
For triangle ABC, AD, CE, and BF are the medians of triangle ABC, D is the midpoint of BC as AD is the median. BC = 10 units, DC = 5 units. Therefore, DC = 5 units.

Example 3: ABC is an equilateral triangle, and AD is the altitude through A. Show that (AD)^{2} = 3(BD)^{2}.
Solution:
Since the triangle is equilateral, the altitude AD is also the median, and so BD = CD = 1/2BC
Consider ΔABD, by the Pythagorean theorem, we have: (AB)^{2}= (AD)^{2} + (BD)^{2}
⇒ (BC)^{2 }= (AD)^{2 }+ (BD)^{2 }(∵AB = BC = AC)
⇒ (2BD)^{2 }= AD^{2 }+ BD^{2 }(∵BD = 12BC)
⇒ 4(BD)^{2 }= (AD)^{2 }+ (BD)^{2 }⇒ (AD)^{2 }= 3(BD)^{2}
Therefore, (AD)^{2 }= 3(BD)^{2}. Hence Proved.
FAQs on Median of a Triangle
Does the Median of a Triangle Divide it into Two Congruent Triangles?
For a given triangle, its median further divides the triangle into two congruent triangles and the median then becomes the common side of both the triangles.
What is the Difference Between the Altitude and Median of a Triangle?
A median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side, dividing it further into two congruent triangles. An altitude of a triangle is the line segment joining a vertex of a triangle with the opposite side such that the segment is perpendicular to the opposite side. It's the height of the triangle.
What are the Properties of the Median of a Triangle?
A median of a triangle is a line segment joining the vertex of the triangle to the midpoint of its opposite side. The median of a triangle bisects the opposite side, dividing it into two equal halves, and bisects the angle from which it arises into two angles of equal measures.
Is Median Always 90 Degrees?
No, the Median doesn't always form a right angle to the side on which it is falling. It is only in the case of an equilateral triangle or isosceles triangle that one median falls on the nonequal side of the isosceles triangle.
What is the Difference Between Median and Bisector?
The median of a triangle refers to a line segment joining a vertex to the midpoint of the opposite side whereas a bisector is referred to as a line segment that passes through the midpoint of a segment, which is also perpendicular to the segment.
What is the Formula of the Median of a Triangle?
The formula for the median of a triangle is given by using Apollonius’s Theorem \(\mathrm{m}_{\mathrm{a}}=\sqrt{\frac{2 b^{2}+2 c^{2}a^{2}}{4}}\) where,
 a, b, c are the sides of the triangle
 ma is the length of the median from vertex A.
How to Find the Equation of the Median of a Triangle?
To find the equation of the median of a triangle we can use the formula, twopoint form equation of the straight line connecting the vertex and the midpoint of the opposite vertex, we get \(\left(\mathbf{y}\mathbf{y}_{1}\right)=\left(\frac{\mathrm{y}_{2}\mathrm{y}_{1}}{\mathrm{x}_{2}\mathrm{x}_{1}}\right)_{\mathrm{AD}}\left(\mathrm{x}\mathrm{x}_{1}\right)\)