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

🏆Practice parts of a triangle

The center of the triangle

  1. 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.
  2. The intersection point of the medians - the centroid - divides each median in a ratio of 2:12:1 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!

einstein

True or false:

DE not a side in any of the triangles.
AAABBBCCCDDDEEE

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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:

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.

In triangle ABCABC shown here, we can observe that the purple point MM represents the intersection point of the three medians in the triangle.
Point MM 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 22 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 ABCABC there are two medians ADAD and BEBE intersecting at point MM.
From this, it follows that if segment CWCW is a median, it must pass through point MM, and conversely, if CECE passes through point MM, we can determine that it is a median to side ABAB
Note: We can determine that if 22 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 ABCABC

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:
CECE is a median in the triangle
BWBW is a median in the triangle
and - ADAD passes through point MM.

It is also known that:

DB=5DB=5
BE=4BE=4
​​​​​​​AW=4​​​​​​​AW=4

  1. Determine CDCD
  2. Determine the perimeter of the triangle

Solution:

  1. We know that ADAD passes through point MM which is the same point where the two medians CECE and BWBW intersect.
    Therefore, according to the theorem that all three medians intersect at one point, we can determine that ADAD is also a median because if 22 medians meet at a certain point, the third median must pass through it as well.
    We are given that DB=5DB=5 therefore CD=5CD=5 given that a median divides the side into two equal parts.
  2. To determine the perimeter of the triangle we must identify all of its sides.

AE=4AE = 4 since CECE is a median
CW=4CW = 4 since BWBW is a median

And we found CDCD in part a.
Therefore:
AB+BC+AC=AB+BC+AC=
8+10+8=268+10+8=26

The perimeter of triangle ABCABC is 2626 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 ABCABC the three medians intersect at point MM.
According to the theorem, point MM divides each median in a ratio of 2:12:1 where the larger part of the median is closer to the vertex.
Thus we can determine that:
AM=2xAM=2x
MD=xMD=x

And:
CM=2YCM=2Y
ME=YME=Y

And:
BM=2ZBM=2Z
MW=ZMW=Z

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

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:
ADAD is a median
BWBW is a median
and CECE passes through point MM

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

Determine CMCM and WMWM
Solution:
Since we are given that: ADAD is a median and BWBW is a median and CECE passes through point MM, we can conclude that CECE 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 2:12:1 where the larger part of the median is closer to the vertex, and given that: ME=2ME=2 (the smaller part), we can conclude that:
CM=4CM= 4
Since BM=5BM=5 is the larger part closer to the vertex, we can conclude that WM=2.5WM=2.5

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Examples with solutions for Parts of a Triangle

Exercise #1

True or false:

DE not a side in any of the triangles.
AAABBBCCCDDDEEE

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?
AAABBBCCCDDDEEE

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

The triangle ABC is shown below.

To which side(s) are the median and the altitude drawn?

AAABBBCCCDDDEEEFFF

Step-by-Step Solution

To solve the problem of identifying to which side of triangle ABC ABC the median and the altitude are drawn, let's analyze the diagram given for triangle ABC ABC .

  • We acknowledge that a median is a line segment drawn from a vertex to the midpoint of the opposite side. An altitude is a line segment drawn from a vertex perpendicular to the opposite side.
  • Upon reviewing the diagram of triangle ABC ABC , line segment AD AD is a reference term. It appears to meet point C C in the middle, suggesting it's a median, but it also forms right angles suggesting it is an altitude.
  • Given the placement and orientation of AD AD , it is perpendicular to line BC BC (the opposite base for the median from A A ). Therefore, this line is both the median and the altitude to side BC BC .

Thus, the side to which both the median and the altitude are drawn is BC.

Therefore, the correct answer to the problem is the side BC BC , corresponding with choice Option 2: BC \text{Option 2: BC} .

Answer

BC

Exercise #4

The triangle ABC is shown below.

Which line segment is the median?

AAABBBCCCDDDEEEFFF

Video Solution

Step-by-Step Solution

To solve this problem, we need to identify the median in triangle ABC:

  • Step 1: Recall the definition of a median. A median is a line segment drawn from a vertex to the midpoint of the opposite side.
  • Step 2: Begin by evaluating each line segment based on the definition.
  • Step 3: Identify points on triangle ABC:
    • AD is from A to a point on BC.
    • BE is from B to a point on AC.
    • FC is from F to a point on AB.
  • Step 4: Determine if these points (D, E, F) are midpoints:
    • Since BE connects B to E, and E is indicated to be the midpoint of segment AC (as shown), BE is the median.
    • AD and FC, by visual inspection, do not connect to midpoints on BC or AB respectively.

Therefore, the line segment that represents the median is BE BE .

Thus, the correct answer is: BE

Answer

BE

Exercise #5

Look at triangle ABC below.

What is the median of the triangle and to which side is it drawn?

AAABBBCCCDDDEEE

Step-by-Step Solution

A median of a triangle is a line segment that connects a vertex to the midpoint of the opposite side. In triangle ABC \triangle ABC , we need to identify such a median from the diagram provided.

Step 1: Observe the diagram to identify the midpoint of each side.

Step 2: It is given that point E E is located on side AC AC . If E E is the midpoint of AC AC , then any line from a vertex to point E E would be a median.

Step 3: Check line segment BE BE . This line runs from vertex B B to point E E .

Step 4: Since E E is labeled as the midpoint of AC AC , line BE BE is the median of ABC \triangle ABC drawn to side AC AC .

Therefore, the median of the triangle is BE BE for AC AC .

Answer

BE for AC

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