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.
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:
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
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:
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
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?
Incorrect
Correct Answer:
Not true
Question 2
The triangle ABC is shown below.
To which side(s) are the median and the altitude drawn?
Incorrect
Correct Answer:
BC
Question 3
The triangle ABC is shown below.
Which line segment is the median?
Incorrect
Correct Answer:
BE
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.
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
The triangle ABC is shown below.
To which side(s) are the median and the altitude drawn?
Step-by-Step Solution
To solve the problem of identifying to which side of triangle ABC the median and the altitude are drawn, let's analyze the diagram given for triangle 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, line segment AD is a reference term. It appears to meet point 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, it is perpendicular to line BC (the opposite base for the median from A). Therefore, this line is both the median and the altitude to side 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, corresponding with choice Option 2: BC.
Answer
BC
Exercise #4
The triangle ABC is shown below.
Which line segment is the median?
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.
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?
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, 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 is located on side AC. If E is the midpoint of AC, then any line from a vertex to point E would be a median.
Step 3: Check line segment BE. This line runs from vertex B to point E.
Step 4: Since E is labeled as the midpoint of AC, line BE is the median of △ABC drawn to side AC.
Therefore, the median of the triangle is BE for AC.