Median of a Triangle

Median of a Triangle
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Let's have a look at Fred's confusion!

Fred made his first pizza.

He is in great joy! He made 4 cuts to it for 4 of his friends.

More friends came in. Now, how will he divide the pizza so that each gets the same amount?

Have you ever noticed, from any interior angle, a cut to the midpoint of the opposite side divides it into congruent halves? 

Suppose if four more friends join and want to try his pizza, Fred can cut those four triangular pieces along their medians, from any interior angle to the midpoint of the opposite side!

The pieces will be smaller, but they will be of equal area.

In this lesson, we will explore the equation of the median of a triangle formula, sum of medians of a triangle, altitude of a triangle, median of equilateral triangle formula, and median of a triangle definition while discovering the interesting facts around them. 

Let's explore how to find medians of a triangle:

Now that you have explored the simulation, stay tuned with us to learn more about the median of a triangle formula, median of a triangle calculator, median of a triangle theorem, median of a triangle equation, the median of a triangle example, and median of triangle properties.

Lesson Plan

What Is Meant by the Median of a Triangle?

We are already familiar with the properties of a triangle, let's have a look at the medians of a triangle.

Median of a Triangle Definition

A line segment, joining a vertex to the mid-point of the side, opposite to that vertex, is called the median of a triangle.

AD is the median of triangle ABC

AD is the median in the above figure, dividing BC into two halves, such that, DB = DC.

The median of a triangle theorem states that the medians of a triangle intersect at a point called the centroid, which is two-thirds of the distance from the vertices to the midpoint of the opposite sides.

Properties of Medians of a Triangle

  • Every triangle has 3 medians, one from each vertex. 
  • The 3 medians always meet at a single point, no matter what the shape of the triangle is.
  • The point where the 3 medians meet is called the centroid of the triangle.
  • Each median of a triangle divides the triangle into two smaller triangles which have equal area.
  • In fact, the 3 medians divide the triangle into 6 smaller triangles of equal area.
  • The sum of two sides of a triangle is greater than the median drawn from the vertex, which is common.
  • The median and lengths of sides:  “3 times the sum of squares of the length of sides = 4 times the squares of medians of a triangle.”

How to Find the Median of a Triangle?

We have seen in the introduction section as how to find median of a triangle

Now, let's have a look on how to calculate the length of each median:

\(m_{a}=\sqrt{\frac{2 b^{2}+2 c^{2}-c^{2}}{4}}\)

\(m_{b}=\sqrt{\frac{2 a^{2}+2 c^{2}-b^{2}}{4}}\)

\(m_{c}=\sqrt{\frac{2 b^{2}+2 c^{2}-a^{2}}{4}}\)

a, b and c = three sides of the triangle.

Now, explore the median of a triangle calculator given below, to find the coordinates of the medians of triangles.

How to Find the Equation of Median of a Triangle?

Step 1: Find the coordinates of the vertex of triangles.

Step 2: Find the coordinates of midpoints of line segments.

Step 3: Join the vertex to the midpoint of the opposite side of the triangle. You will get medians.

Step 4: Find the equation of the median of a triangle.

 
Thinking out of the box
Think Tank
  1. A sculpture is planning to make a new sculpture, that consists of a triangle balanced on another triangle. Which point could be the point of balance? Does finding centroid would help?

More About Median of a Triangle:

  1. The perimeter of a triangle is greater than the sum of its three medians.
  2. The median of a triangle divides the triangle into two triangles with equal areas.

AD is the median of triangle ABC, h = height for two triangles formed by median AD

The two triangles ABD  and ADC have the same area, with equal heights and same base lengths(BD=DC, as D, is the midpoint of BC since AD is the median of the triangle)

3. For a given triangle, the second median divides the triangle formed by the first median in the ratio 1:2

(Let AD and BE, be two medians)

AD and EC are two medians of triangle ABC

4. For a given triangle, its three medians divide it further into six triangles, all equal in area.

Triangle with three medians

All 6 triangles AOE, AOF, BOE, BOD, COD, OFC, AOF, are equal in area.

5. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid.

  • Median of an equilateral triangle:
In an equilateral triangle, each altitude, median and angle bisector drawn from the same vertex, overlap. Also, the length of the medians is equal.
Median of equilateral triangle formula: \(m_{a}=m_{8}=m_{6}=a \frac{\sqrt{3}}{2}\)
  • Medians of an isosceles triangle:
The medians from the vertices having equal angles are of equal length in the case of an isosceles triangle.
  • Median of a scalene triangle:
The length of the medians differs in a scalene triangle.
 
What about the medians of acute, obtuse, and right-angled triangles?

Triangle Centers

Centroid
The centroid of a triangle is the center of the triangle, which can be determined as the point of intersection of all the three medians of a triangle. The centroid divides each median into two parts, which are always in the ratio 2:1. The centroid formula is used to determine the coordinates of a triangle’s centroid. 
Circumcenter
The point, which is equidistant from all the vertices of the triangle.
Incenter
The point, which is equidistant from the sides of the triangle.
Orthocenter
The point at which all the altitudes of the triangle intersect.

Congruence and Similarity in Triangles

The corresponding sides, perimeters, medians and altitudes, all will be in the same ratio, for two similar triangles

The midpoint of a segment divides it into two congruent segments, whereas the median of a triangle divides it into two congruent triangles.

Also, if the two triangles are congruent. The medians of congruent triangles are equal as corresponding parts of congruent triangles are congruent.


Solving Triangles

"Solving Triangles" means finding missing sides and angles. Here are the properties of the triangle that should be taken care of while solving triangles:

  • The Angle Sum Property states that the sum of the three interior angles of a triangle is 180°
  • The Triangle Inequality Property sum of the length of the two sides of a triangle is greater than the third side.
  • The Exterior Angle Property of a triangle is always equal to the sum of the interior opposite angles. 
  • In the Congruence Property, two triangles are said to be congruent if all their corresponding sides and angles are congruent.
  • The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is the sum of the squares of the other two sides.
  • The sum of two sides of a triangle is greater than the median drawn from the vertex, which is common.

Have a look at triangle calculation, to know more about solving triangles.

 
important notes to remember
Important Notes
  1. Each median divides the triangle into two smaller triangles which have the same area
  2. The centroid (the point where they meet) is the center of gravity of the triangle
  3. The three medians divide the triangle into 6 smaller triangles that all have the same area, even though they may have different shapes.
  4. Sum of medians of a triangle: The sum of the squares of the medians of a triangle equals three-fourths the sum of the squares of the sides of the triangle.

Solved Examples

Example 1

 

 

Give a term that describes the point O, shown in the figure given below.

Triangle with three medians and center point O

Solution

The point O is the center of the triangle and the point of intersection of all the three medians of a triangle.

Thus, the given point O is Centroid.

\(\therefore\) O is the centroid.
Example 2

 

 

For the given triangle ABC, G is the centroid and BC = 10 units. Determine the length of BD.

Triangle with three medians and centroid G

Solution

For triangle ABC,

AD, CE, and BF are the medians of triangle ABC,

D is the midpoint of BC as AD is the median.

BC = 10 units, DC = 5 units.

\(\therefore\) DC = 5 units.
Example 3

 

 

\( ABC\) is an equilateral triangle, and \(AD\) is the altitude through \(A\).

Show that \(A{D^2} = 3B{D^2}\).

Solution

Since the triangle is equilateral, the altitude \(AD\) is also the median, and so

\[BD = CD = \frac{1}{2}BC\]

Consider \(\Delta ABD\), by the Pythagorean theorem, we have:

\[\begin{align}
  A{B^2} &= A{D^2} + B{D^2} \hfill \\
\Rightarrow\! B{C^2}\! &=\!\! A{D^2}\! +\! B{D^2} \! \left ( {\because \!AB\!\! =\! BC\!\! =\! AC\!} \right) \hfill\\
   \Rightarrow \!{\left(\! {2BD} \right)^2} \!&= \!A{D^2}\! +\! B{D^2} \!\left(\! {\because\! BD\! =\! \!\frac{1}{2}BC}\! \right) \hfill \\
   \Rightarrow\! 4B{D^2}\! &=\! A{D^2}\! +\! B{D^2} \hfill \\
   \Rightarrow A{D^2} &= 3B{D^2} \hfill \\ 
\end{align} \]

\(\therefore\) \(\ A{D^2} = 3B{D^2} \) 

Interactive Questions

Here are a few activities for you to practice.

Select/Type your answer and click the "Check Answer" button to see the result.

 

 
 
 
 

Let's Summarize

We hope you enjoyed learning about the median of a triangle, median of a triangle formula, sum of medians of a triangle, altitude of a triangle, median of equilateral triangle formula, and median of a triangle definition with the simulations and practice questions while discovering the interesting facts around them.

Now you will be able to easily solve problems related to these and also about the median of a triangle formula, median of a triangle calculator, median of a triangle theorem, median of a triangle equation, the median of a triangle example, and median of triangle properties.

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Frequently Asked Questions (FAQs)

1. Does the median of a triangle divides it into two congruent triangles?

For a given triangle, its median further divides the triangle into two congruent triangles.

2. What is the difference between the altitude and median of a triangle?

A median of a triangle is a line segment that joins its vertex to its mid-point of the opposite side, dividing it further, into two congruent triangles.

An altitude of a triangle is the line segment joining a vertex of a triangle with the opposite side such that the segment is perpendicular to the opposite side. It's the height of the triangle.

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