The Centroid of a triangle

The centroid of a triangle is the point of intersection of the three medians of that triangle.

In the following acute triangle \(ABC\),
\(D\), \(E\) and \(F\) are the midpoints of \(AB\), \(BC\) and \(CA\).
So, \(CD\), \(AE\) and \(BF\) are the medians of this triangle.
The medians intersect at the point \(G\).
This point \(G\) is the centroid of \(△ABC\).

What is a median?

Note: A median is a segment that is drawn from a vertex of a triangle to the mid-point of the opposite side of that vertex. Each median of a triangle divides the triangle into two smaller triangles that have equal areas.
A triangle has three medians because it has three sides. The three medians divide the triangle into 6 smaller triangles of equal area.

Properties of a Centroid

• ✓ Centroid is the point of intersection of the medians.
• ✓ It is the geometric center of an object.
• ✓ It always lies inside the triangle.
• ✓ A centroid divides the medians in \(2:1\) ratio.
• ✓ In an equilateral triangle, the centroid, the orthocenter,
•  the circumcenter and the incenter coincide on the same point.

Centroid of an Acute triangle

Centroid of an Obtuse triangle

Centroid of a Right triangle

Centroid of an Equilateral triangle

Centroid of an Isosceles triangle

Coordinates of centroid

If coordinates of the vertices \(A\), \(B\) and \(C\) are \( (x_1, y_1) \), \( (x_2, y_2) \) and \( (x_3, y_3) \) respectively, then the coordinates of the centroid \(G\) can be calculated with the following formula: \[ G(x, y) = \left( \frac{x_1+x_2+x_3}{3}, \frac{y_1+y_2+y_3}{3}\right) \] For a triangle with vertices \(A(-30, 3)\), \(B(0, 3)\) and \(C(10, 30)\), the coordinates of the centroid \(G\) may be calculated as follows:
Here, \(x_1=-30,\; x_2=0,\; x_3=10,\; y_1=3,\; y_2=3,\; y_3=30 \)

Coordinates of the centroid \(G\): \begin{align} G(x, y) &= \left( \frac{-30 + 0 + 10}{3}, \frac{3 + 3 + 30}{3}\right)\\\\ &= \left( \frac{-20}{3}, \frac{36}{3}\right) \\\\ &= ( -6.67, 12 ) \\\\ \end{align} So the coordinates of the centroid of the above triangle is \(( -6.67, 12 )\)

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