Formula of Median of Triangle

Formula for length of first median (m_a) of a triangle, where the median is formed on side ‘a’, is given by:

[Tex]m_a~=~\frac{1}{2} \sqrt{2b^2 + 2c^2 – a^2}[/Tex]

For any triangle ABC if its sides AB, AC and BC are given then its formula is calculated as,

Formula to calculates the length of the median from vertex A to the midpoint of side BC in a triangle ABC, where sides ( a ), ( b), and (c) represent the lengths of the sides of the triangle opposite vertices A, B, and C, respectively.

For example, consider a triangle ABC where (AB = 5), (AC = 6), and (BC = 7). To find the length of median m_a from vertex A to the midpoint of side BC:

m_a = [Tex]\frac{1}{2} \sqrt{2(6)^2 + 2(7)^2 – (5)^2}[/Tex]

m_a = [Tex]\frac{1}{2} \sqrt{72 + 98 – 25}[/Tex]

m_a = 1/2 √145

m_a = 1/2 × 12.04

m_a ≈ 6.02

So, length of first median (m_a) in this triangle is approximately 6.02 units.

Similar formulas and calculations can be used for the second median (m_b) and third median (m_c) of the triangle, formed on sides ‘b’ and ‘c’ respectively.

How to Find Median of Triangle with Coordinates?

To find the median of a triangle with coordinates, you can follow these steps:

Step 1: Identify Coordinates: First, identify coordinates of vertices of triangle. Let’s denote them as (x₁​, y₁​), (x₂​, y₂​), and (x₃​, y₃​).

Step 2: Calculate Midpoint of Opposite Side: Choose one of the sides of triangle as opposite side for which you want to find median. Calculate midpoint of this side using midpoint formula: Midpoint = (x₁​ + x₂)/2​​, (y₁ ​+ y₂​​)/2

Step 3: Use Distance Formula to Find Length: Once you have the midpoint of the opposite side, use the distance formula to find the length of the median from the vertex to this midpoint: [Tex]d = \sqrt{(x_2 – x_1)^2 + (y_2 – y_1)^2}[/Tex]

Step 4: Repeat for Other Vertices: Repeat steps 2 and 3 for other two sides of triangle to find lengths of other two medians.

Step 5: Determine Median: Compare lengths of three medians obtained. Median with shortest length is median of triangle.

Length of Median Formula

Formula to calculate the length (m) of a median of a triangle depends on the lengths of the sides of the triangle. If (a), (b), and (c) represent the lengths of the sides of triangle, and the median is formed on side ‘a’, then the formula for the length of the median is given by:

• [Tex]m_a = \frac{1}{2} \sqrt{2b^2 + 2c^2 – a^2}[/Tex]

Similarly, for the medians formed on sides ‘b’ and ‘c’, the formulas are:

• [Tex]m_b = \frac{1}{2} \sqrt{2a^2 + 2c^2 – b^2}[/Tex]
• [Tex]m_c = \frac{1}{2} \sqrt{2a^2 + 2b^2 – c^2}[/Tex]

Median of a Triangle is a line segment that joins a vertex of a triangle to the midpoint of the opposite side. A median divides the joining into two equal parts. Each triangle has three medians, one originating from each vertex. These medians intersect at a point called the centroid, which lies within the triangle.

In this article, we will learn about, Median of Triangle Definition, Properties of Median of Triangle, Examples related to Median of Triangle, and others in detail.