How to Find the Centroid and Medians of a Triangle

Centroid and median of triangle

A median is a line that goes from one corner of a triangle to the midpoint on the opposite side.

Theory

The Centroid and Medians

The medians have one common point of intersection. This point is called the centroid $G$ . The centroid $G$ divides all the medians in the ratio $2 : 1$ . This results in the following relationships:

\frac{A G}{G P} = \frac{B G}{G Q} = \frac{C G}{G R} = 2

When you want to find the centroid of a triangle, you need to draw two of the medians. You draw the medians by drawing a straight line from every corner to the midpoint on the opposite side. The intersection of these lines is the centroid.

Example 1

A triangle $△ A B C$ has the sides $A B = 6$ , $A C = 4$ and $B C = 7$ . Construct the centroid of the triangle.

Before you construct the centroid $G$ , you need to construct the triangle with the given measures. Start with the line $A B = 6$ . Set the compass’s radius to 7 and make a faint circle with center $B$ . Then, set the compass’s radius to 4 and make a faint circle with center $A$ . The corner $C$ appears as either of the intersection points between the two circles. Then you end up with the following triangle:

Example of construction of centroid of triangle 1

Then you construct the angle bisector for two of the sides. At the intersection between these, you have the incenter, which you call $I$ . Then you find the midpoint to two of the sides and draw a straight line from each of these points to the opposite corner. The centroid $G$ of the triangle is the intersection between these two lines:

Example of construction of centroid of triangle 2

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