Median of a Triangle

The median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. Every triangle has exactly three medians. Each median splits the triangle into two smaller triangles of equal area. Median helps in understanding a triangle's balance and symmetry. It is essential for solving problems related to area, centroid, and triangle properties. In this guide, you will learn about the definition of the median of a triangle, its properties, and related examples in simple terms.

Table of Contents

• What is the Median of a Triangle
• Properties of the Median of a Triangle
• Solved Examples on the Median of a Triangle

What is the Median of a Triangle

The median of a triangle is a line segment that joins a vertex to the midpoint of the opposite side. It connects one corner of a triangle to the middle of the opposite side.

Consider ∆ABC. Let D be the midpoint of BC. Then AD is a median of the triangle.

Properties of the Median of a Triangle

• Every triangle has three medians: from A to the midpoint of BC, from B to the midpoint of AC, and from C to the midpoint of AB.

• Each median divides the triangle into two equal-area triangles.

• The three medians are concurrent at the centroid G.

• The centroid divides the median in the ratio 2:1.

• The three medians divide the triangle into six smaller triangles of equal area.

• Length of the median: For a triangle with sides of lengths a, b, and c.

        The median to side a is (1/2)√(2b² + 2c² − a²). 

       Similarly: Median to side b = (1/2)√(2a² + 2c² - b²) and Median to side c = (1/2)√(2a² + 2b² - c²)

• In an equilateral triangle, all medians are equal. Also, Median = altitude = angle bisector

• In an isosceles triangle, two medians are equal. Medians corresponding to equal sides of an isosceles triangle are equal.

• In a scalene triangle, all medians are different.
  
  

Solved examples on the Median of a Triangle

Example 1: Find the length of the median to side a if: b = 6 cm, c = 8 cm, a = 10 cm.
Solution:
Length of  Median = (1/2) √(2b² + 2c² − a²)
= (1/2) √(2(6²) + 2(8²) − 10²)
= (1/2) √(2×36 + 2×64 − 100)
= (1/2) √(72 + 128 − 100)
= (1/2) √100
= 5 cm

Example 2: Find the length of the median PM of triangle PQR, where the sides are PQ = 10 units, PR = 13 units, and QR = 8 units. Here, PM is the median drawn to side QR
Solution: Let a = PQ = 10 units, b = PR = 13 units, and c = QR = 8 units.

PM is the median drawn to side QR.

Length of Median = (1/2)√(2a² + 2b² - c²)

= (1/2) √(2(10²) + 2(13²) − 8²)
= (1/2) √(2×100 + 2×169 − 64)
= (1/2) √(200 + 338 − 64)
= (1/2) √474
= 10.885 units.

Example 3: In triangle ABC, median AD = 9 cm. Find the length of AG, where G is the centroid.
Solution: Given, median AD = 9 cm. The centroid divides the median in the ratio 2:1.
Length of AG = (⅔) × AD = (⅔) × 9 = 6 cm