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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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

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

Question 2

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

Question 3

Can a plane angle be found in a triangle?

Examples with solutions for Parts of a Triangle

Exercise #1

Can a triangle have a right angle?

Video Solution

Step-by-Step Solution

To determine if a triangle can have a right angle, consider the following explanation:

Definition of a Right Angle: An angle is classified as a right angle if it measures exactly $90^\circ$ .
Definition of a Right Triangle: A right triangle is a type of triangle that contains exactly one right angle.
According to the definition, a right triangle specifically includes a right angle. This is a well-established classification of triangles in geometry.

Thus, a triangle can indeed have a right angle and is referred to as a right triangle.

Therefore, the solution to the problem is Yes.

Answer

Yes

Exercise #2

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

Video Solution

Step-by-Step Solution

The triangle's altitude is a line drawn from a vertex perpendicular to the opposite side. The vertical line in the diagram extends from the triangle's top vertex straight down to its base. By definition of altitude, this line is the height if it forms a right angle with the base.

To solve this problem, we'll verify that the line in question satisfies the altitude condition:

Step 1: Identify the triangle's vertices and base. From the diagram, the base appears horizontal, and the vertex lies directly above it.
Step 2: Check the nature of the line. The line is vertical when the base is horizontal, indicating perpendicularity.
Conclusion: The vertical line forms right angles with the base, thus acting as the altitude or height.

Therefore, the straight line depicted is indeed the height of the triangle. The answer is Yes.

Answer

Yes

Exercise #3

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

Video Solution

Step-by-Step Solution

To determine if the straight line is the height of the triangle, we'll analyze its role within the triangle:

Step 1: Observe the triangle and the given line. The triangle seems to be made of three sides and a vertical line within it.
Step 2: Recall that the height of a triangle, in geometry, is defined as a perpendicular dropped from a vertex to the opposite side.
Step 3: Examine the positioning of the line: The vertical line starts at one vertex of the triangle and intersects the base, appearing to be perpendicular.
Step 4: Verify perpendicularity: Given that the line is shown as a clear vertical (and a small perpendicular indicator suggests perpendicularity), we accept this line as the height.
Step 5: Conclude with verification that the line is effectively meeting the definition of height for the triangle in the diagram.

Therefore, the vertical line in the figure is indeed the height of the triangle.

Yes

Answer

Yes

Exercise #4

Can a plane angle be found in a triangle?

Video Solution

Step-by-Step Solution

To determine whether a plane angle can be found in a triangle, we need to understand what a plane angle is and compare it to the angles within a triangle.

A plane angle is an angle formed by two lines lying in the same plane.
In the context of geometry, angles found within a triangle are the interior angles, which are the angles between the sides of the triangle.
Although the angles in a triangle are indeed contained within a plane (since a triangle itself is a planar figure), when referencing "plane angles" in geometry, we usually consider angles related to different geometric configurations beyond those specifically internal to defined planar shapes like a triangle.
The term "plane angle" typically refers to the measurement of an angle in radians or degrees within a plane, but this doesn't specifically pertain to angles of a triangle.

Therefore, based on the context and usual geometric conventions, the concept of a "plane angle" is not typically used to describe the angles found within a triangle. Thus, a plane angle as defined generally in geometry is not found specifically within a triangle.

Therefore, the correct answer to the question is $\text{No}$ .

Answer

Exercise #5

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

Video Solution

Step-by-Step Solution

In the given problem, we have a triangle depicted with a specific line drawn inside it. The question asks if this line represents the height of the triangle. To resolve this question, we need to discern whether the line is perpendicular to one of the sides of the triangle when extended, as only a line that is perpendicular from a vertex to its opposite side can be considered the height.

The line in question is shown intersecting one of the sides within the triangle but does not form a perpendicular angle with any side shown or the ground (as is required for it to be the height of the triangle). A proper height would typically intersect perpendicularly either at or along the extended line of the opposite side from a vertex.

Therefore, based on the visual clues provided and the typical geometric definition of a height (or altitude) in a triangle, this specific line does not fit the criteria for being a height.

Thus, we conclude that the line depicted is not the height of the triangle. The correct answer is No.

Answer

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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