### 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**:

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

**Step 1***: Draw a line segment \(AB = 8\) \(cm\) using the ruler.*

**Step 2***: With \(A\) as centre, mark an angle \(55^{\circ}\) using the protractor and name it as \(X\). Now join \(AX\).*

**Step 3***: With \(B\) as centre, measure \(6\) \(cm\) in the compass and cut an arc intersecting \(AX\) and mark it as \(C\).*

**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.*

**Step 7***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**:

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

**Step 1***: Draw a line segment \(AB = 5\) \(cm\) using the ruler.*

**Step 2***: With \(B\) and \(A\) as centre, cut arcs of radius \(6\) \(cm\) and \(7\) \(cm\) respectively to meet each other at \(C\).*

**Step 3***: 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 4***: Mark the intersecting point of the perpendicular bisectors and the sides \(AB\) and \(BC\) as \(P\) and \(Q\), respectively.*

**Step 5***: Draw the medians \(AQ\) and \(CP\) to meet each other at \(G\), which is the centroid of the triangle.*

**Step 6**