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.

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Test yourself on parts of a triangle!

Is DE side in one 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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Question 1

Determine the type of angle given.

Question 2

Determine the type of angle given.

Question 3

Is the straight line in the figure the height of the triangle?

Examples with solutions for Parts of a Triangle

Exercise #1

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 #2

Determine the type of angle given.

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

Step 1: Examine the diagram presented.
Step 2: Identify any familiar angle formations or configurations.
Step 3: Use knowledge of angles to classify the type shown.
Step 4: Determine the correct response from available options.

Observing the diagram:

The diagram includes two lines, one horizontal and the other vertical, extending fully. This horizontal extent along with the linear continuation suggests it forms an angle at the intersection with $180^\circ$ . This indicates a straight angle.

We classify straight angles because an angle formed by two lines directly facing opposite directions is known to measure $180^\circ$ . This diagrammatic representation aligns perfectly to confirm it calculates and visually shows a straight angle.

Thus, by recognizing these details within the diagram, we confirm the type of angle as Straight.

Answer

Right

Exercise #3

Determine the type of angle given.

Video Solution

Step-by-Step Solution

The problem involves classifying the angle represented visually, which looks like a semicircle with a central axis drawn. This indicates an angle that spans half a complete circle.

A complete circle measures $360^\circ$ , so half of it, represented by a semicircle, measures half of $360^\circ$ , which is $180^\circ$ .

The four primary classifications for angles are:

Acute: Less than $90^\circ$
Right: Exactly $90^\circ$
Obtuse: Greater than $90^\circ$ but less than $180^\circ$
Straight: Exactly $180^\circ$

Since the angle measures exactly $180^\circ$ , it is classified as a straight angle.

Therefore, the type of angle given is Straight.

Answer

Straight

Exercise #4

Is the straight line in the figure the height of the triangle?

Video Solution

Step-by-Step Solution

The task is to determine whether the line shown in the diagram serves as the height of the triangle. For a line to be considered the height (or altitude) of a triangle, it needs to be a perpendicular segment from a vertex to the line that contains the opposite side, often referred to as the base.

Let's analyze the diagram:

The triangle is described by its vertices, forming a shape, and one side is the base. There's a line drawn from one vertex directed toward the opposite side.
To be the height, this line must be perpendicular to the side it meets (the base).
Though the figure does not explicitly show perpendicularity with a right angle mark, the line appears as a straight, direct connection from the vertex to the base. This is typically indicative of it being a height.
Assuming typical geometric conventions and the common depiction of heights in diagrams, the line shows properties consistent with being perpendicular to the opposite side, thereby functioning as the height.

Based on the analysis, the line is indeed the height of the triangle. Thus, the answer is Yes.

Therefore, the solution to the problem is Yes.

Answer

Yes

Exercise #5

Is the straight line in the figure the height of the triangle?

Video Solution

Step-by-Step Solution

To determine if the straight line in the figure is the height of the triangle, we must verify the following:

The line segment must extend from a vertex of the triangle and be perpendicular to the opposite side (or its extension).

In examining the figure provided, we notice that the triangle is formed by vertices at points $A, B,$ and $C$ . Let's assume the base is the line segment $\overline{BC}$ .

The line in question extends from a vertex $A$ and appears to intersect the base $BC$ at a right angle.

Since it is extending from vertex to the opposite side and forming a right angle with it, this line meets the definition of an altitude.

Therefore, the line in the figure is indeed the height of the triangle. By confirming the perpendicular relationship, we determine that this geometric feature correctly describes an altitude.

Yes, the straight line in the figure is the height of the triangle.

Answer

Yes

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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