### Theory:

Centroid: It is the point of concurrency of a triangle formed by the intersection of its medians. It is denoted by $$G$$.
Working rule to construct the Centroid of a Triangle
Case 1:

Given the length of two sides of a triangle and the measure of one of its interior angle.
Example:
Construct a triangle $$ABC$$ given $$AB = 8$$ $$cm$$, $$BC = 6$$ $$cm$$ and $$\angle BAC = 55^{\circ}$$ and locate its centroid.

Construction:

Step 1: Draw a rough figure for the given question to get the picture of the triangle that is to be constructed.

Step 2: Draw a line segment $$AB = 8$$ $$cm$$ using the ruler.

Step 3: With $$A$$ as centre, mark an angle $$55^{\circ}$$ using the protractor and name it as $$X$$. Now join $$AX$$.

Step 4: With $$B$$ as centre, measure $$6$$ $$cm$$ in the compass and cut an arc intersecting $$AX$$ and mark it as $$C$$.

Step 5: Draw the perpendicular bisectors of any two sides of the triangle $$AB$$ and $$BC$$ (say) to find the mid-points of the sides, respectively.

Step 6: Mark the intersecting point of the perpendicular bisectors and the sides $$AB$$ and $$BC$$ as $$P$$ and $$Q$$ respectively.

Step 7: Draw the medians $$AQ$$ and $$CP$$ to meet each other at $$G$$, which is the centroid of the triangle.

Case 2:

Given the length of all the three sides of a triangle.
Example:
Construct a triangle $$ABC$$ given $$AB = 5$$ $$cm$$, $$BC = 6$$ $$cm$$ and $$CA = 7$$ $$cm$$ and locate its centroid.

Construction:

Step 1: Draw a rough figure for the given question to get the picture of the triangle that is to be constructed.

Step 2: Draw a line segment $$AB = 5$$ $$cm$$ using the ruler.

Step 3: With $$B$$ and $$A$$ as centre, cut arcs of radius $$6$$ $$cm$$ and $$7$$ $$cm$$ respectively to meet each other at $$C$$.

Step 4: Draw the perpendicular bisectors of any two sides of the triangle $$AB$$ and $$BC$$ (say) to find the mid-points of the sides, respectively.

Step 5: Mark the intersecting point of the perpendicular bisectors and the sides $$AB$$ and $$BC$$ as $$P$$ and $$Q$$, respectively.

Step 6: Draw the medians $$AQ$$ and $$CP$$ to meet each other at $$G$$, which is the centroid of the triangle.