### Theory:

Incentre: It is the point of concurrency of a triangle formed by the intersection of its three angle bisectors of a triangle. It is denoted by \(I\).

Incircle:It is the circle drawn inside the triangle with incentre as its centre.

*Working rule to construct the Incircle 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 the incircle of a triangle \(ABC\) given \(AB = 8\) \(cm\), \(BC = 6\) \(cm\) and \(\angle BAC = 55^{\circ}\). Locate its incentre and find the radius.

**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 angle bisectors of any two angles of the triangle \(A\) and \(B\) (say).*

**Step 5***: Mark the intersecting point of the angle bisectors as \(I\), which is the incentre of the given triangle.*

**Step 6***: Draw a perpendicular bisector from \(I\) to one of the sides of the triangle \(AB\) (say) to meet \(AB\) at \(D\).*

**Step 7***: Measure the length of \(ID\) to get the inradius of the incircle.*

**Step 8***: With \(I\) as centre and \(ID\) as radius, draw the incircle of the triangle.*

**Step 9***Case 2*:

Given the length of a side of a triangle and the measure of two of its interior angle.

Example:

Construct the incircle of a triangle \(ABC\) given \(AB = 6\) \(cm\), \(\angle BAC = 55^{\circ}\) and \(\angle CBA = 60^{\circ}\). Locate its incentre and find the radius.

**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 \(AB = 6\) \(cm\) 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, mark an angle \(60^{\circ}\) using the protractor and name it as \(Y\). Now join \(BY\).*

**Step 4***: Mark the intersecting point of \(AX\) and \(BY\) as \(C\).*

**Step 5***: Draw the angle bisectors of any two angles of the triangle \(A\) and \(B\) (say).*

**Step 6***: Mark the intersecting point of the angle bisectors as \(I\), which is the incentre of the given triangle.*

**Step 7***: Draw a perpendicular bisector from \(I\) to one of the sides of the triangle \(AB\) (say) to meet \(AB\) at \(D\).*

**Step 8***: Measure the length of \(ID\) to get the inradius of the incircle.*

**Step 9***: With \(I\) as centre and \(ID\) as radius draw the incircle of the triangle.*

**Step 10**