Median in a triangle

A median in a triangle is a line segment that extends from a vertex to the midpoint of the opposite side, dividing it into two equal parts.

Additional properties:

In every triangle, it is possible to draw 3 medians.
All 3 medians intersect at one point.
The median to a side in a triangle creates 2 triangles of equal area.
In an equilateral triangle - the median is also a height and an angle bisector.
In an isosceles triangle - the median from the vertex angle is also a height and an angle bisector.
In a right triangle - the median to the hypotenuse equals half the hypotenuse.

Diagram of triangle ABC showing medians AD, BE, and CF intersecting at the centroid. The centroid divides each median into a 2:1 ratio, illustrating triangle centroid properties.

Detailed explanation

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Test yourself on parts of a triangle!

True or false:

DE not a side in any of the triangles.

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Median in a triangle

In this article, we will learn everything you need to know about medians in a triangle! Don't worry, the material about medians in a triangle is both easy and straightforward to understand.

Define a median in a triangle?

A median in a triangle is a line segment that extends from a vertex to the midpoint of the opposite side, dividing it into two equal parts.
Remember that "median" in real life represents the middle point, and similarly here it divides the side in the middle!

We can observe this in the following drawing:

Diagram of a triangle ABC with a median AD drawn from vertex A to the midpoint D of side BC, illustrating geometric properties of medians.

In triangle $ABC$
$AD$ is a median - it extends from the vertex $A$ and divides the opposite side $CB$ into two
equal parts: $CD=BD$

Additional Properties of a Median in a Triangle:

In every triangle, it is possible to draw 3 medians.
All 3 medians intersect at one point.
The median to a side of a triangle creates 2 triangles of equal area.

You can observe this below:

Since there are 3 vertices in a triangle, there can be 3 medians.
Each median extends from a vertex to the opposite side and bisects it.
All medians intersect at one point.

Reminder:
How do we calculate the area of a triangle?

$height*corresponding~side~to~height \over 2$

If we take for example the triangle $ABC$ and want to calculate its area when:
$AD$ height = $6$
$BC = 8$

Diagram of triangle ABC with median AD labeled. The length of median AD is shown as 6 units, and segment BD is labeled as 4 units, highlighting the geometric properties of a triangle's median.

We can deduce that the area of triangle $ABC$ is:
$\frac{6*8}{2}=24$
Now if we draw the median $AD$ we can observe that the two triangles it creates are equal in area.
The side is divided in the middle thus it is identical in both triangles and the height is identical.
Therefore, the area of each created triangle is identical and will be equal to half the area of triangle $ABC$

$\frac{6*4}{2}=12$

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Test your knowledge

Question 1

Is DE side in one of the triangles?

Question 2

True or false:

AB is a side of the triangle ABC.

Question 3

True or false:

AD is a side of triangle ABC.

3 important statements about medians:

In an equilateral triangle - the median is also a height and an angle bisector.
As shown in the figure:

$AD$ is a height to side $CB$
and also a median to side $CB$ (divides it into two equal parts)
as well as an angle bisector

Diagram of an isosceles triangle with a median intersecting the base at a right angle. The median forms a perpendicular bisector, and the top angle is highlighted in purple.

$A$

In an isosceles triangle - the median drawn from the vertex angle is also a height and an angle bisector.
Let's observe this in the figure below:

In an isosceles triangle - the median drawn from the vertex angle is also a height and an angle bisector.

$AD$ is a median drawn from vertex angle $A$ .
It is also a height to side $CB$ , as well as a median to $CB$ , in addition to bisecting the vertex angle $A$ .

3. In a right triangle - the median to the hypotenuse equals half the hypotenuse.

We can observe this in the figure below:

Triangle $ABC$ is a right triangle.
$CD$ is the median to the hypotenuse and equals half of the hypotenuse.
That is
$CD=AD=DB$

Exercise:

Given:

$CD$ is a median in triangle $ABC$
$CE$ is a median in triangle $ACD$

$AB=14$

Diagram of a triangle with medians AD and CE drawn in orange and purple, intersecting at a single point inside the triangle. The vertices are labeled A, B, and C, while the midpoints of the sides are labeled D and E.

Calculate the length of segment $AE$ .
Determine the area of triangle $ECD$ if it is known that the area of triangle $ACE$ is $6$ ?
Based on your answer in part b, determine the area of triangle $DCB$

Solution:

Let's mark the given information in the drawing.

Diagram of a triangle labeled A, B, and C, with medians AD (orange) and CE (purple). The length of CE is marked as 3.5. A green arc labeled 14 spans from vertex A to vertex B, highlighting the angle subtended by side AB.

Given that – $AB =14$
Since $CD$ is a median,
$AD=DB=7$
due to the fact that the median bisects the side at its midpoint.
Given that $CE$ is also a median.
Therefore $AE=ED=3.5$ .

2. Given that the area of triangle $ACE$ is $6$
The area of triangle $ECD$
must also be $6$ . A median divides the triangle into two triangles of equal area.

3. The area of triangle $DCB$ must be equal to the area of triangle $ACD$ .

Triangle $ACD$ consists of two triangles with equal areas that sum up to $12$ .
Therefore, the area of triangle $DCB$ is $12$ .

Do you know what the answer is?

Question 1

True or false:

BC is a side of triangle ABC.

Question 2

ABC is an isosceles triangle.

AD is the median.

What is the size of angle $∢\text{ADC}$ ?

Question 3

AD is the median in triangle ABC.

BD = 4

Find the length of DC.

Examples with solutions for Parts of a Triangle

Exercise #1

True or false:

DE not a side in any of the triangles.

Video Solution

Step-by-Step Solution

To solve the problem of determining whether DE is not a side in any of the triangles, we will methodically identify the triangles present in the diagram and examine their sides:

Identify triangles in the diagram. The diagram presented forms a right-angled triangle ABC with additional lines forming smaller triangles within.
Triangles formed: Triangle ABC (major triangle), Triangle ABD, Triangle BEC, and Triangle DBE.
Let's examine the sides of these triangles:

Triangle ABC has sides AB, BC, and CA.
Triangle ABD has sides AB, BD, and DA.
Triangle BEC has sides BE, EC, and CB.
Triangle DBE has sides DB, BE, and ED.

Notice that while point D is used, the segment DE is only part of line BE and isn't listed as a direct side of any triangle.

Therefore, the claim that DE is not a side in any of the triangles is indeed correct.

Hence, the answer is True.

Answer

True

Exercise #2

Is DE side in one of the triangles?

Video Solution

Step-by-Step Solution

Since line segment DE does not correspond to a full side of any of the triangles present within the given geometry, we conclude that the statement “DE is a side in one of the triangles” is Not true.

Answer

Not true

Exercise #3

True or false:

AB is a side of the triangle ABC.

Video Solution

Step-by-Step Solution

To solve this problem, let's clarify the role of AB in the context of triangle ABC by analyzing its diagram:

Step 1: Identify the vertices of the triangle. According to the diagram, the vertices of the triangle are points labeled A, B, and C.
Step 2: Determine the sides of the triangle. In any triangle, the sides are the segments connecting pairs of distinct vertices.
Step 3: Identify AB as a line segment connecting vertex A and vertex B, labeled directly in the diagram.

Considering these steps, line segment AB connects vertex A with vertex B, and hence, forms one of the sides of the triangle ABC. Therefore, AB is indeed a side of triangle ABC as shown in the diagram.

The conclusion here is solidly supported by our observation of the given triangle. Thus, the statement that AB is a side of the triangle ABC is True.

Answer

True

Exercise #4

True or false:

AD is a side of triangle ABC.

Video Solution

Step-by-Step Solution

To determine if line segment AD is a side of triangle ABC, we need to agree on the definition of a triangle's side. A triangle consists of three sides, each connecting pairs of its vertices. In triangle ABC, these sides are AB, BC, and CA. Each side is composed of a direct line segment connecting the listed vertices.

In the diagram provided, there is no indication of a point D connected to point A or any other vertex of triangle ABC. To claim AD as a side, D would need to be one of the vertices B or C, or a commonly recognized point forming part of the triangle’s defined structure. The provided figure and description do not support that AD exists within the given triangle framework, as no point D is defined within or connecting any existing vertices.

Therefore, according to the problem's context and based on the definition of the sides of a triangle, AD cannot be considered a side of triangle ABC. It follows that the statement "AD is a side of triangle ABC" should be deemed not true.

Answer

Not true

Exercise #5

True or false:

BC is a side of triangle ABC.

Video Solution

Step-by-Step Solution

To solve this problem, we must determine whether BC is indeed a side of triangle ABC. A triangle consists of three vertices connected by three line segments that form its sides.

Firstly, observe the triangle labeled in the diagram with vertices A, B, and C. For triangle ABC, the sides are composed of the segments that connect these points.

The three line segments connecting the vertices are:
- $AB$ , connecting points A and B;
- $BC$ , connecting points B and C; and
- $CA$ , connecting points C and A.

Among these, BC is clearly listed as one of the segments connecting two vertices of the triangle. Therefore, BC is indeed a side of triangle ABC.

Hence, the statement is True.

Answer

True

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Median in a triangle

Median in a triangle

Test yourself on parts of a triangle!

Median in a triangle

Define a median in a triangle?

3 important statements about medians:

Exercise:

Examples with solutions for Parts of a Triangle

Exercise #1

Video Solution

Step-by-Step Solution

Answer

Exercise #2

Video Solution

Step-by-Step Solution

Answer

Exercise #3

Video Solution

Step-by-Step Solution

Answer

Exercise #4

Video Solution

Step-by-Step Solution

Answer

Exercise #5

Video Solution

Step-by-Step Solution

Answer

More Questions

Parts of a Triangle