Median of a Triangle (original) (raw)

Last Updated : 23 Jul, 2025

**Median of a Triangle is a line segment that joins a vertex of a triangle to the midpoint of the opposite side. It plays a crucial role in understanding the properties and balance of triangles. A median divides the joining into two equal parts. Each triangle has three medians, one originating from each vertex and intersecting at the center of the triangle.

In this article, we will learn about the **median of a triangle definition, the properties of a median of a triangle, examples related to the median of a triangle, and others in detail.

Table of Content

**Median of a triangle is a line segment that connects one vertex of the triangle to the midpoint of the opposite side. In other words, it divides the opposite side into two equal parts. For example, in the given figure where AD is the median, it connects vertex A to the midpoint of side BC, splitting BC into two equal segments BD and DC. This characteristic holds for all triangles, regardless of their size or shape.

Median-of-Triangle

Median of a triangle plays a significant role in geometry, helping to identify important properties and relationships within the triangle. These medians intersect at a point called the centroid, which lies within the triangle.

Median of Triangle Definition

Median of a triangle is a line segment that joins one vertex to midpoint of opposite side, dividing side into two equal parts. Three medians in a triangle, each originating from a vertex and intersecting at the centroid, the triangle's center of mass.

Properties of the median of a triangle are:

Median-of-a-Triangle

Both median and altitude have different purposes,

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.

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

m_a~=~\frac{1}{2} \sqrt{2b^2 + 2c^2 - a^2}

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

Median-of-Triangle-Formula

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:

ma = \frac{1}{2} \sqrt{2(6)^2 + 2(7)^2 - (5)^2}

ma = \frac{1}{2} \sqrt{72 + 98 - 25}

ma = 1/2 √145

ma = 1/2 × 12.04

ma ≈ 6.02

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

Similar formulas and calculations can be used for the second median (mb) and third median (mc) of the triangle, formed on sides 'b' and 'c' respectively.

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

**Step 1: Identify Coordinates: First, identify coordinates of vertices of triangle. Let's denote them as (__x_1​, __y_1​), (__x_2​, __y_2​), and (__x_3​, __y_3​).

**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 = ( x**1 ​ + **x**2 )/2​​, ( y**1 **​+ **y**2 ​​)/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: d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

**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:

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

We can construct the median of any triangle using following steps:

**Step 1: Sketch the triangle on paper, labelling its vertices A, B, and C.

**Step 2: Locate the midpoint of each side by drawing perpendicular lines from the midpoint of each side.

**Step 3: Draw lines from each midpoint to the opposite vertex to form the medians.

Median can be drawn to any kind of triangle, such as:

Median of Equilateral Triangle

In an equilateral triangle, the median possesses distinct characteristics. The median of an equilateral triangle is a line segment that connects a vertex to the midpoint of the opposite side, bisecting it. Since all sides of an equilateral triangle are equal, the medians from each vertex are also equal in length.

Additionally, all three medians in an equilateral triangle coincide at a single point, known as the centroid. This centroid divides each median in a ratio of 2:1, with the longer segment closer to the vertex. The median of an equilateral triangle is a crucial element in understanding the symmetrical properties and balance within this particular type of triangle.

The formula to find the length of the median of an equilateral triangle depends on the length of its sides.

If ' s' represents the length of each side of an equilateral triangle, then the length '**m' of the median is calculated using the formula:

**m **= √3/2 **s

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Some examples on Median of a Triangle are,

**Example 1: In triangle DEF, if DE = 7 cm, DF = 9 cm, and EF = 10 cm, calculate the length of the median from vertex D to side EF.

**Solution:

To find length of median from vertex D to side EF in triangle DEF, we can use formula for length of a median in terms of lengths of sides of triangle. Let's denote length of median as (m).

Given,

Use Formula:

m2 = \frac{2(DF^2 + DE^2) - EF^2}{4}

Put given values:

m2 = \frac{2(9^2 + 7^2) - 10^2}{4}

⇒ m2 = \frac{2(81 + 49) - 100}{4}

⇒ m2 = \frac{2(130) - 100}{4}

⇒ m2 = (260 - 100)/4

⇒ m2 = 160/40

⇒ m2 = 40

⇒ m = √40

⇒ m = 2√10

So, length of median from vertex D to side EF is 2√10 cm.

**Example 2: Determine the length of median from vertex C to side AB in a triangle where AC = 12 cm, BC = 9 cm, and AB = 15 cm.

**Solution:

To find length of median from vertex C to side AB in given triangle, use formula for length of median:

m2 = \frac{2(BC^2 + AC^2) - AB^2}{4}

Given,

Use Formula:

m2 = \frac{2(9^2 + 12^2) - 15^2}{4}

⇒ m2 = \frac{2(81 + 144) - 225}{4}

⇒ m2 = \frac{2(225) - 225}{4}

⇒ m2 = \frac{450 - 225}{4}

⇒ m2 = 225/4

⇒ m = \sqrt{\frac{225}{4}}

⇒ m = 15/2

⇒ m = 7.5

So, length of median from vertex C to side AB is (7.5) cm

**Example 3: Triangle UVW has vertices U (3, 5), V (9, 5), and W (6, 1). Find length of median of triangle with the coordinates of vertices from vertex U to side VW .

**Solution:

To find length of median from vertex U to side VW in triangle UVW, First determine midpoint of side VW, and then use distance formula to find length of median.

Given,

**1. Find midpoint of side VW

Midpoint formula is given by:

\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)

Using coordinates of V and W:

\text{Midpoint}(VW) = \left( \frac{9 + 6}{2}, \frac{5 + 1}{2} \right)

=~\left( \frac{15}{2}, \frac{6}{2} \right)

=~\left( \frac{15}{2}, 3 \right)

So, the midpoint of side VW is \left( \frac{15}{2}, 3 \right)

**2. Now, use distance formula to find length of median from vertex U to midpoint of side VW

Distance Formula is:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Given,

\text{Midpoint}(VW) = \left( \frac{15}{2}, 3 \right)

Using distance formula:

d = \sqrt{\left( \frac{15}{2} - 3 \right)^2 + (3 - 5)^2}

⇒ d = \sqrt{\left( \frac{15}{2} - 3 \right)^2 + (-2)^2}

⇒ d = \sqrt{\left( \frac{15}{2} - \frac{6}{2} \right)^2 + 4}

⇒ d = \sqrt{\left( \frac{9}{2} \right)^2 + 4}

⇒ d = \sqrt{\frac{81}{4} + 4}

⇒ d = \sqrt{\frac{81 + 16}{4}}

⇒ d = \sqrt{\frac{97}{4}}

⇒ d = √(97)/2

So, length of median from vertex U to side VW is √(97)/2

**Example 4 : Given the vertices of a triangle A(2,3), B(8,7), and C(5,12). find the lengths of all three median of triangle with the coordinates of vertices.

**Solution:

To find length of all medians in triangle, First determine midpoint of the opposite sides , and then use distance formula to find length of median.

Given,

1. Find the midpoints of each side:

2. Calculate the lengths of the medians:

The length of medians are **7.91 units, **4.53 units and **7 units.

Some practice questions regarding the Median of a triangle are,

**Q1: In triangle ABC, if AB = 8 cm and AC = 6 cm, find the length of the median from vertex A to side BC.

**Q2: Triangle XYZ has vertices X (1, 2), Y (4, 6), and Z (7, 2). Calculate the length of the median of a triangle with the coordinates of vertices from vertex X to side YZ.

**Q3: Determine the length of the median from vertex B to side AC in a triangle where AB = 10 cm, BC = 12 cm, and AC = 14 cm.

**Q4: Given triangle PQR with PQ = 15 cm, PR = 20 cm, and QR = 25 cm, find the length of the median from vertex P to side QR.

**Q5: Triangle LMN has vertices L(2, 4), M(6, 8), and N(8, 2). Find the length of the median of triangle with the coordinates of vertices from vertex L to side MN.

Conclusion

Median of a triangle is an important line that helps us understand triangles better. Median of triangle divides the triangle into two equal areas and is key in finding the triangle's centroid, or center of balance where all three medians of a triangle intersect, it serves as the triangle's center of gravity and plays a crucial role in many applications. Knowing how to find and use medians makes solving geometric problems easier. Overall, the median is a valuable concept in both learning and applying geometry.