Median of a Triangle

The 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. All triangles have exactly three medians, one from each vertex. These medians intersect each other at the triangle's centroid. Let us learn more about what is the median of a triangle, the median of triangle formula, and the properties of median of triangle in this article.

The median of a triangle is a line segment that joins one vertex to the midpoint of the opposite side of a triangle. Observe the figure given below, in which AD is the median, dividing BC into two equal parts, such that, BD = DC.

Median of a Triangle Definition

A line segment, joining a vertex to the mid-point of the side opposite to that vertex, is called the median of a triangle.

The median of a triangle can be easily identified with the help of the following properties:

• The median of a triangle is a line segment joining the vertex of the triangle to the mid-point of its opposite side.
• It bisects the opposite side, dividing it into two equal parts.
• The median of a triangle further divides the triangle into two triangles having the same area.
• Irrespective of the shape or size of a triangle, its three medians meet at a single point.
• Every triangle has 3 medians, one from each vertex. The point of concurrency of 3 medians forms the centroid of the triangle.
• 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.

Altitude and Median of Triangle

The altitude and the median of a triangle are different from each other. The median of a triangle is defined as the line segment that joins the vertex and the mid-point of the opposite side of the triangle. All triangles have 3 medians (one from each vertex), meeting at a single point, irrespective of the type of the triangle. The 3 medians are located inside the triangle and they meet at a common point called the centroid of the triangle. A median always bisects the opposite side on which it is formed.

The altitude of a triangle is defined as a line segment joining the vertex to the opposite side of the triangle at a right angle (90°). An altitude can be located inside or outside a triangle depending on the type of triangle. All triangles have 3 altitudes (one from each vertex), meeting at a single point of the triangle known as the Orthocenter. The orthocenter may lie inside or outside the triangle. An altitude may not necessarily bisect the opposite side on which it is formed.

How to Find the Median of a Triangle?

The median of a triangle can be calculated using a basic formula that applies to all three medians. Let us learn the formula that is used to calculate the length of each median.

Median of Triangle Formula

The formula for the first median of a triangle is as follows, where the median of the triangle is m_a, the sides of the triangle are a, b, c, and the median is formed on side 'a'.

\(m_{a}=\sqrt{\frac{2 b^{2}+2 c^{2}-a^{2}}{4}}\)

The formula for the second median of a triangle is as follows, where the median of the triangle is m_b, the sides of the triangle are a, b, c, and the median is formed on side 'b'.

\(m_{b}=\sqrt{\frac{2 a^{2}+2 c^{2}-b^{2}}{4}}\)

The formula for the third median of a triangle is as follows, where the median of the triangle is m_c, the sides of the triangle are a, b, c, and the median is formed on side 'c'.

\(m_{c}=\sqrt{\frac{2 a^{2}+2 b^{2}-c^{2}}{4}}\)

Let us understand this with the help of an example.

Example: Find the length of the median of the given triangle PQR whose sides are given as follows, PQ = 10 units, PR = 13 units, and QR = 8 units, respectively, in which PM is the median formed on side QR.

Solution: In order to find the length of the median (PM) which is formed on side QR, we will use the median of triangle formula, PM = \(\sqrt{\frac{2 PQ^{2}+2 PR^{2}-QR^{2}}{4}}\); where QR = 8, PQ = 10, PR = 13

Substituting the values in the formula, we will get, Median PM = \(\sqrt{\frac{(2 × 10^{2}) + (2 × 13^{2}) - 8^{2}}{4}}\) = 10.88 units

Now, let us learn how to calculate the length of the median when the coordinates of the triangle are given.

How to find the Median of Triangle with Coordinates?

When the coordinates of the three vertices of a triangle are given, we can follow the steps given below to find the length of the median of the triangle.

Length of Median Formula

Let us see the formula that is used to find the length of median of a triangle when the coordinates of the vertices are given.

• Step 1: Using the coordinates of the vertices of the triangle, find the coordinates of the midpoint of the line segment on which the median is formed. This can be done using the midpoint formula. The formula for midpoint is, [(x₁ + x₂)/2, (y₁ + y₂)/2], where (x₁, y₁) and (x₁, y₁) are the coordinates of the line segment.
• Step 2: After the coordinates of the midpoint are obtained, find the length of the median using the distance formula, where one endpoint is the vertex from where the median starts and the other is the mid point of the line segment on which the median is formed.
• Step 3: The length of the median can be calculated with the distance formula, D = √[(x₂ – x₁)² + (y₂ – y₁)²]; where (x₁, y₁) and (x₁, y₁) are the coordinates of the median.

Let us understand this with the help of an example.

Example: Find the length of the median AD if the coordinates of the triangle ABC are given as, A (4, 10), B (8, 2), C (-8, 4)

Solution: Using the steps given above, we will first find the coordinates of the point D which is the midpoint of side BC on which the median is formed.

• The coordinates of point D using the midpoint formula will be, D = [(x₁ + x₂)/2, (y₁ + y₂)/2], where the coordinates of B = (8, 2) and C = (-8, 4).
• Substituting the values in the formula, we get, [8 +(-8)]/2 and (2 + 4)/2.
• This gives the coordinates of D as (0, 3). Now, the length of the median can be calculated using the distance formula, AD = √[(x₂ – x₁)² + (y₂ – y₁)²]; where the coordinates of the median are A (4, 10), and D (0, 3).
• Substituting the values in the formula, AD = √[(0 - 4)² + (3 - 10)]². This can be solved as √(16 + 49) = 8.06 units. This gives the length of the median AD as 8.06 units.

Median of Equilateral Triangle

In an equilateral triangle, the medians are of equal length. Another property of the median of an equilateral triangle is that the median is the same as the altitude of the equilateral triangle. This is because we know that the perpendicular bisector divides the opposite side of a triangle into two equal parts and we know that a median also has the same property.

Tips on Median of a Triangle

• 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.
• The perimeter of a triangle is greater than the sum of its three medians.
• If the two triangles are congruent, the medians of congruent triangles are equal since the corresponding parts of congruent triangles are congruent.

☛ Related Articles

• What is a Median in a Triangle?
• What is Centroid of A Triangle
• Centroid Formula
• Circumcenter of Triangle
• Incenter of a Triangle
• Orthocenter

What is the Median of a Triangle in Math?

The 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. All triangles have exactly three medians, one from each vertex.

What is the Difference Between the Altitude and Median of a Triangle?

The median of a triangle is defined as the line segment that joins the vertex and the mid-point of the opposite side of the triangle. All triangles have 3 medians (one from each vertex), meeting at a single point, irrespective of the type of the triangle. The 3 medians are located inside the triangle and they meet at a common point called the centroid of the triangle.

The altitude of a triangle is defined as a line segment joining the vertex to the opposite side of the triangle at a right angle (90°). An altitude can be located inside or outside a triangle depending on the type of triangle. It is also known as the height of the triangle. All triangles have 3 altitudes (one from each vertex), meeting at a single point of the triangle known as the Orthocenter. The orthocenter may lie inside or outside the triangle.

How to Find the Median of a Triangle with Sides?

The length of the median of a triangle can be calculated if the length of the three sides is given. The basic formula that is used to calculate the median is, \(m_{a}=\sqrt{\frac{2 b^{2}+2 c^{2}-a^{2}}{4}}\); where the median of the triangle is m_a, the sides of the triangle are a, b, c, and the median is formed on side 'a'.

How to Find the Median of Triangle with Coordinates?

When the coordinates of the triangle are given, the length of the median can be calculated using the following steps:

• Step 1: Using the coordinates of the vertices of the triangle, find the coordinates of the midpoint of the line segment on which the median is formed. This can be done using the midpoint formula. The formula for midpoint is, [(x₁ + x₂)/2, (y₁ + y₂)/2], where (x₁, y₁) and (x₁, y₁) are the coordinates of the line segment.
• Step 2: After the coordinates of the midpoint are obtained, find the length of the median using the distance formula, where one endpoint is the vertex from where the median starts and the other is the mid point of the line segment on which the median is formed.
• Step 3: The length of the median can be calculated with the distance formula, D = √[(x₂ – x₁)² + (y₂ – y₁)²]; where (x₁, y₁) and (x₁, y₁) are the coordinates of the median. A detailed example is given above on this page.

What are the Properties of the Median of a Triangle?

The median of a triangle is a line segment joining the vertex of the triangle to the mid-point of its opposite side. The median of a triangle bisects the opposite side, dividing it into two equal parts. Every triangle has 3 medians, one from each vertex. The point of concurrency of 3 medians is called the centroid of the triangle.

Is the Median of a Triangle 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 in which the median is the same as the altitude; or in the case of an isosceles triangle where the median falls on the non-equal side of the isosceles triangle at an angle of 90°.

What is the Median of Triangle Formula?

The median of triangle formula is given by using Apollonius’ 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.
• m_a is the median formed on side 'a' of the triangle.

Is the Median of a Triangle always the Perpendicular Bisector?

No, the median of a triangle may not always be the perpendicular bisector of the line segment on which it is formed. However, in case of an equilateral triangle, the median is the perpendicular bisector of the line segment.

Is Median of a Triangle an Angle Bisector?

The median of a triangle is an angle bisector if it is an equilateral triangle or if the median starts from the non-congruent angle of an isosceles triangle.

Does the Median of a Triangle Divide it into Two Congruent Triangles?

Yes, the median of a triangle divides it into two congruent triangles and the median then becomes the common side of both the triangles.

What is the Difference Between Median and Perpendicular Bisector?

The median of a triangle refers to a line segment joining a vertex to the mid-point of the opposite side, whereas, a perpendicular bisector is referred to as a line that bisects a line segment at 90°.