Medians and Centroid of a Triangle

Q: How many medians does a triangle have?

Every triangle has exactly three medians, one drawn from each vertex to the midpoint of the opposite side. These three medians always meet at a single point inside the triangle, called the centroid. No triangle has more or fewer than three medians.

Q: Is the centroid always inside a triangle?

Yes. Regardless of the triangle's shape, the centroid always lies inside the triangle. It is the single point where all three medians intersect. Each median runs from a vertex to the opposite side, so their meeting point always stays inside the triangle.

Q: How does the centroid divide each median?

The centroid divides each median into two segments in a 2:1 ratio. The segment from the vertex to the centroid is twice the segment from the centroid to the midpoint. So the longer part always sits nearer the vertex, and the shorter part nearer the opposite side.

Q: How do you find a full side length from its midpoint?

A midpoint splits a side into two equal halves. So the full side is twice the length from the vertex to that midpoint. For example, if E is the midpoint of AB, then AB is twice AE. Just double the half you know to find the complete side.

Key concept

A median of a triangle joins a vertex to the midpoint of the opposite side. The three medians meet at a single point called the centroid, which divides each median in a 2:1 ratio, with the longer part next to the vertex.

Medians and Centroid of a Triangle - introduction visual

Video Lesson

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$Medians and Centroid of a Triangle poster$

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Flashcards

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The definition of a median of a triangle is that it connects a vertex to the midpoint of the opposite side, dividing it into two equal-area triangles.

The definition of the centroid of a triangle is the point where the three medians intersect, dividing each median in a 2:1 ratio.

The median and centroid of a triangle apply to the problem.

Median of a Triangle

A median joins a vertex to the midpoint of the opposite side.
A median splits a triangle into two smaller triangles with equal area.

Practice Questions

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Q1Easy

AD is a median of triangle ABC. If BD $= 5 cm$ , what is the length of CD?

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Interactive Activity

Explore how the centroid divides the median.

Click any dashed median to activate it

Drag A

Given

AG=10

GD=?

Point G is the Centroid

The centroid divides the median in a 2:1 ratio.

Solution

GD=½×AG=5

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Students Also Ask

The questions students bump into most on this topic

How many medians does a triangle have?

Every triangle has exactly three medians, one drawn from each vertex to the midpoint of the opposite side. These three medians always meet at a single point inside the triangle, called the centroid. No triangle has more or fewer than three medians.

Does a median divide a triangle into two equal areas?

Yes. A median divides a triangle into two smaller triangles of equal area. Because the median meets the midpoint of the opposite side, each smaller triangle has exactly half the original area. This holds for every median in any triangle.

Is the centroid always inside a triangle?

Yes. Regardless of the triangle's shape, the centroid always lies inside the triangle. It is the single point where all three medians intersect. Each median runs from a vertex to the opposite side, so their meeting point always stays inside the triangle.

How does the centroid divide each median?

The centroid divides each median into two segments in a 2:1 ratio. The segment from the vertex to the centroid is twice the segment from the centroid to the midpoint. So the longer part always sits nearer the vertex, and the shorter part nearer the opposite side.

How do you find a full side length from its midpoint?

A midpoint splits a side into two equal halves. So the full side is twice the length from the vertex to that midpoint. For example, if E is the midpoint of AB, then AB is twice AE. Just double the half you know to find the complete side.

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