Medians and Centroid of a Triangle
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.
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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.
Centroid of a Triangle
- The centroid is the point where the three medians meet.
- It divides each median in the ratio 2 : 1, measured from the vertex.
Using Medians and the Centroid
- A median always goes to the midpoint of the opposite side.
- The centroid splits each median in a 2 : 1 ratio, with the longer part next to the vertex.
Practice Questions
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AD is a median of triangle ABC. If BD , what is the length of CD?
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A median joins a vertex to the midpoint of the opposite side. Here AD is a median, so D is the midpoint of BC and the two halves are equal: BD = CD.
Since BD , CD . (The diagram is not drawn to scale, so CD only looks longer.)
What is the name of the point where the medians of a triangle intersect?
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The three medians of a triangle always meet at one point, called the centroid. It always lies inside the triangle, whatever the triangle's shape, and it is the triangle's balance point.
The other names belong to different points: the circumcentre is where the perpendicular bisectors of the sides meet, the incentre is where the angle bisectors meet, and the orthocentre is where the altitudes (perpendicular heights) meet.
If AD is a median and E is the centroid of triangle ABC, and AE , what is the length of ED?
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The centroid divides every median in the ratio , measured from the vertex. Here that means , so ED is half of AE.
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If CG is a median and E is the centroid of triangle ABC, and the area of triangle AGC is , what is the area of triangle ABC?
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Because CG is a median, G is the midpoint of AB, so . Triangles AGC and BGC have equal bases and share the same perpendicular height from C, so they have equal areas.
Triangle AGC is therefore exactly half of triangle ABC:
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The centroid E is not needed for this question.
If BF is a median and E is the centroid of triangle ABC, and BF , what is the length of BE?
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The centroid divides the median in a 2:1 ratio. The longer segment BE is two-thirds of the total length BF: .
If E is the centroid of triangle ABC, and you know that AE , what is the length of the median from A to BC?
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The centroid divides each median in the ratio from the vertex, so AE is of the whole median AD.
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Check: the shorter piece is cm, and cm.
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Interactive Activity
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Frequently asked questions
The questions students bump into most on this topic
A median is a line from a vertex to the midpoint of the opposite side. Every triangle has three, one from each corner. A median splits the triangle into two smaller triangles of equal area, so if one half has an area of 10 cm², the whole triangle is 20 cm². The three medians always cross at one point, the centroid.
The centroid is the single point where a triangle’s three medians meet. Whatever the triangle’s shape, it always lies inside. It divides each median in the ratio 2 : 1, with the longer part next to the vertex. For a triangle cut from card, the centroid is also the balance point, which is why it is sometimes called the centre of gravity.
Draw two medians: find the midpoint of a side, join it to the opposite vertex, and repeat for a second side. The point where the two medians cross is the centroid; the third median passes through it too, which is a good check. You do not need all three unless you want that check.
In the ratio 2 : 1: the part from the vertex to the centroid is twice the part from the centroid to the midpoint, so it is two thirds of the median and the other part is one third. If AD is 6 and D is the centroid, then DG is 3. If DE is 4, then CD is 8 and the whole median CE is 12.
Double the half you know. A median meets the opposite side at its midpoint, so the two halves are equal: if E is the midpoint of AB and AE is 5, then AB is 10. In the same way, if BD is 5 on the median AD, then DC is 5 and BC is 10.
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