Median Divides a Triangle into Two Triangles of Equal Area

A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. One of its key properties is that it divides the triangle into two smaller triangles of equal area. This fundamental concept in geometry helps in understanding area relationships and solving related problems with ease. In this guide, you will learn about the theorem, its proof and related examples, helping you solve problems more quickly and confidently. 

Table of Contents

• What is a median
• Median Divides a Triangle into Two Triangles of Equal Area: Proof
• Solved Examples on Theorem

What is a median?

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.

Median Divides the Triangle into Two Triangles of Equal Area: Proof

Statement: Each median divides the triangle into two triangles of equal area.

Proof: Let ABC be a triangle. Let AD be a median of ∆ABC. 

∴ BD = DC = 1/2 × BC  --------- (1)

To prove that ar(ABD) = ar(ADC)

Construction: Draw line AN ⊥ BC.

In  ∆ABD, 

ar(∆ABD) = 1/2 × base × height = 1/2 × BD × AN  --------- (2)

In  ∆ACD, 

ar(∆ACD) = 1/2 × base × height = 1/2 × DC × AN

ar(∆ACD) = 1/2 × BD × AN (∵ D is the midpoint of BC, BD = DC)  ---------- (3)

From (2) and (3)

ar(∆ABD) = ar(∆ACD)

Hence proved.

 

Solved Examples on Theorem

Example 1: In triangle ABC, AD is a median. If the area of triangle ABC is 20 cm², find the area of triangle ABD.

Solution: Given, ar(∆ABC) =  20 cm²

The median AD divides ∆ABC into two equal triangles. ∴  ar(∆ABD) = (1/2) × ar(∆ABC) = (1/2) × 20 = 10 cm².

Example 2: The area of triangle ABD is 15 cm², and AD is a median. Find the area of triangle ABC.

Solution: Given ar(∆ABD) = 15 cm². AD is a median.

The median AD divides ∆ABC into two equal triangles.

 ∴  ar(∆ABC) = 2 × ar(∆ABD) = 2 × 15 = 30 cm².