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.
Table of Content
What is Median of a Triangle?
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 a triangle plays a significant role in geometry, helping to identify important properties and relationships within the triangle.
Definition of Median of Triangle
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 Median of Triangle
Properties of the median of a triangle are:
- Median of a triangle bisects the opposite side, dividing it into two equal segments.
- Each triangle has exactly three medians, one from each vertex.
- Medians of a triangle intersect at a single point called the centroid, which is the triangle’s center of mass.
- Centroid divides each median in a ratio of 2:1, with the longer segment closer to the midpoint of the opposite side.
- Centroid of a triangle is also the centroid of its three vertices, meaning it’s the balance point if the triangle were made of a uniform material.
Altitude and Median of Triangle
- Median of a triangle is a line segment that connects one vertex to the midpoint of the opposite side, dividing that side into two equal parts. It helps in finding centroid, which is center of mass of triangle.
- Altitude of a triangle is a perpendicular line segment from a vertex to the opposite side or its extension. It represents height of triangle and is crucial in determining area of triangle.
Both median and altitude have different purposes,
- Medians focus on division and centroid determination.
- Altitudes emphasize height and area calculation.
Formula of Median of Triangle
Formula for length of first median (ma) of a triangle, where the median is formed on side ‘a’, is given by:
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 ma from vertex A to the midpoint of side BC:
ma =
ma =
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.
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:
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 lengths of 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:
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 length of each side of equilateral triangle, then length ‘m’ of median is calculated using the formula:
m = √3/2 s
Read More,
Examples on Median of a Triangle
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,
- DE = 7
- DF = 9
- EF = 10
Use Formula:
m2 =
Put given values:
m2 =
m2 =
m2 =
m2 = (260 – 100)/4
m2 = 160/40
m2 = 40
Taking square root of both sides to find (m)
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 =
Given,
- AC = 12
- BC = 9
- AB = 15
Use Formula:
m2 =
m2 =
m2 =
m2 =
m2 = 225/4
Taking square root of both sides to find (m)
m =
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 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,
- Coordinates of Vertex = V(9, 5)
- Coordinates of Vertex = W(6, 1)
1. Find midpoint of side VW
Midpoint formula is given by:
Using coordinates of V and W:
So, the midpoint of side VW is
2. Now, use distance formula to find length of median from vertex U to midpoint of side VW
Distance Formula is:
Given,
- Coordinates of Vertex = U(3, 5)
Using distance formula:
d =
d =
d =
d =
d =
d =
d =
d = √(97)/2
So, length of median from vertex U to side VW is √(97)/2
Practice Questions on Median of a Triangle
Some practice questions regarding 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 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 from vertex L to side MN.
Median of a Triangle FAQs
What is Median of Triangle Class 9?
In geometry, a median of a triangle is a line segment that connects a vertex of the triangle to the midpoint of the opposite side. It divides the opposite side into two equal segments and plays a significant role in determining the centroid of the triangle.
How Many Medians does a Triangle Have?
A triangle has exactly three medians, one from each vertex. Each median originates from a vertex and connects to midpoint of opposite side, effectively dividing side into two equal parts.
Are Median and Altitude of Triangle Same?
No, median and altitude of a triangle are not same. While both are line segments originating from a vertex and intersecting the opposite side, they serve different purposes. A median divides opposite side into two equal parts, while an altitude is perpendicular to opposite side, representing height of triangle.
How to Find Median of a Triangle with Sides?
To find length of a median of a triangle with known side lengths, one can use formula involving lengths of sides. Depending on which side median is formed on, appropriate formula can be used to calculate its length.
How to Find Median of Triangle with Coordinates?
To find length of a median of a triangle with known coordinates of vertices, one can use midpoint formula to find midpoint of opposite side, then apply distance formula to find length of median.
What are Properties of Median of a Triangle?
- Median divides the opposite side into two equal segments.
- Each triangle has exactly three medians, one from each vertex.
- Medians of a triangle intersect at a single point called the centroid.
Is Median of a Triangle Always 90 degree?
No, median does not always are 90 degrees.
What is Median of Triangle Formula?
Formula to find length of a median of triangle with sides ‘a’ is,