Center of a Triangle - The Intersection Point of Medians

The center of the triangle

All three medians in a triangle intersect at a single point called the centroid -
If two medians intersect at a point inside the triangle, the third median must pass through it as well.
The intersection point of the medians - the centroid - divides each median in a ratio of $2:1$ where the larger part of the median is closer to the vertex.

Diagram of a rectangle labeled ABCD with a marked midpoint M at the intersection of its diagonals. The rectangle is black with white and orange highlights, showcasing symmetry and geometry 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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Center of a triangle - the intersection point of the medians

The center point of a triangle is also called the intersection point of medians or the meeting point of medians.
Remember - a median is a line segment that extends from a vertex to the opposite side and divides it exactly in half.
This can be observed in the following illustration:

In triangle $ABC$ shown here, we can observe that the purple point $M$ represents the intersection point of the three medians in the triangle.
Point $M$ is also the centroid of the triangle.
Important theorems regarding the intersection point and the medians in a triangle:

All three medians in a triangle intersect at one point called the centroid of the triangle.

The theorem states that if $2$ medians intersect at a certain point, then the third median in the triangle must also pass through the same point and intersect at that point, which is called the centroid.

Let's look at an example:

Diagram of a rectangle labeled ABCD with diagonals intersecting at point M, representing the centroid. Additional markings include equal parts on sides and angles labeled for geometric properties. Black rectangle with orange and blue highlights for clarity.

In triangle $ABC$ there are two medians $AD$ and $BE$ intersecting at point $M$ .
From this, it follows that if segment $CW$ is a median, it must pass through point $M$ , and conversely, if $CE$ passes through point $M$ , we can determine that it is a median to side $AB$
Note: We can determine that if $2$ medians in a triangle intersect at a certain point, it will be the centroid.

Let's practice the first theorem about the centroid:
Here is triangle $ABC$

Diagram of a rectangle labeled ABCD with diagonals intersecting at point M, representing the centroid. Additional labels include measurements of 4 units on sides and 5 units on the base, highlighting geometric properties. Black rectangle with orange details for emphasis.

Given that:
$CE$ is a median in the triangle
$BW$ is a median in the triangle
and - $AD$ passes through point $M$ .

It is also known that:

$DB=5$
$BE=4$
$AW=4$

Determine $CD$
Determine the perimeter of the triangle

Solution:

We know that $AD$ passes through point $M$ which is the same point where the two medians $CE$ and $BW$ intersect.
Therefore, according to the theorem that all three medians intersect at one point, we can determine that $AD$ is also a median because if $2$ medians meet at a certain point, the third median must pass through it as well.
We are given that $DB=5$ therefore $CD=5$ given that a median divides the side into two equal parts.
To determine the perimeter of the triangle we must identify all of its sides.

$AE = 4$ since $CE$ is a median
$CW = 4$ since $BW$ is a median

And we found $CD$ in part a.
Therefore:
$AB+BC+AC=$
$8+10+8=26$

The perimeter of triangle $ABC$ is $26$ cm.

The intersection point of the medians - the centroid - divides each median in a ratio of $2:1$ where the larger part of the median is closer to the vertex.

Let's look at an example:

Geometric diagram of a rectangle labeled ABCD with diagonals intersecting at point M (centroid). Variables are labeled: 2X, X, 2Y, Y, Z, and 2Z, illustrating proportional relationships. Black background with orange and white text for clarity.

In triangle $ABC$ the three medians intersect at point $M$ .
According to the theorem, point $M$ divides each median in a ratio of $2:1$ where the larger part of the median is closer to the vertex.
Thus we can determine that:
$AM=2x$
$MD=x$

And:
$CM=2Y$
$ME=Y$

And:
$BM=2Z$
$MW=Z$

Now we will practice the second theorem about the centroid:
Here is triangle $ABC$

eometric diagram of a rectangle labeled ABCD with diagonals intersecting at point M (centroid). Points W, M, and E are marked along the diagonals to illustrate geometric properties. Black background with orange and white text for clarity.

Given that:
$AD$ is a median
$BW$ is a median
and $CE$ passes through point $M$

It is also given that: $ME=2$
and $BM=5$

Determine $CM$ and $WM$
Solution:
Since we are given that: $AD$ is a median and $BW$ is a median and $CE$ passes through point $M$ , we can conclude that $CE$ is a median because if two medians intersect at a certain point, the third median must pass through it.
According to the second theorem which states that the intersection point of the medians divides each median in a ratio of $2:1$ where the larger part of the median is closer to the vertex, and given that: $ME=2$ (the smaller part), we can conclude that:
$CM= 4$
Since $BM=5$ is the larger part closer to the vertex, we can conclude that $WM=2.5$

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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.

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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Center of a Triangle - The Centroid - The Intersection Point of Medians

The center of the triangle

Test yourself on parts of a triangle!

Center of a triangle - the intersection point of the medians

All three medians in a triangle intersect at one point called the centroid of the triangle.

The intersection point of the medians - the centroid - divides each median in a ratio of \(2:1\) where the larger part of the median is closer to the vertex.

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