**Math Labs with Activity – Incircle of a given Triangle by Paper Folding Method**

**OBJECTIVE**

To draw the incircle of a given triangle by the method of paper folding

**Materials Required**

- A sheet of white paper
- A geometry box

**Theory**

The point of intersection of the internal bisectors of the angles of a triangle gives the incentre of the triangle.

Let I be the incentre of a ΔABC. We drop a perpendicular from I on the side BC. Let ID ⊥ BC.

Taking I as the centre and ID as the radius, we can draw the incircle of the ΔABC (see Figure 32.1).

**Procedure**

**Step 1:** Draw any triangle on the sheet of white paper.

Mark its vertices as A, B and C. We shall draw the incircle of the ΔABC.

**Step 2:** Fold the paper along the line passing through the vertex A such that the side AB falls over the side AC. Make a crease and unfold the paper.

Draw a line AX along the crease. Then, AX is the internal bisector of ∠A as shown in Figure 32.2.

**Step 3:** Fold the paper along the line passing through the vertex B such that the side BC falls over the side AB. Make a crease and unfold the paper. Draw a line BY along the crease. Then, BY is the internal bisector of ∠B as shown in Figure 32.3.

**Step 4:** Mark the point of intersection of the two angle bisectors as the point I.

**Step 5:** Fold the paper along the line that passes through the point I and cuts the line BC in such a way that one part of the line BC falls over the other part. Make a crease and unfold the paper. Mark a point D where the line of fold cuts the line BC. Join ID as shown in Figure 32.4.

**Step 6:** Taking / as the centre and ID as the radius, we draw a circle (using a pair of compasses) as in Figure 32.4.

**Result**

The circle drawn in Figure 32.4 is the incircle of the given triangle.

**Remarks:** The teacher must explain it to the students that since all the angle bisectors of a triangle meet at a point, it is sufficient to construct only two angle bisectors so as to obtain their point of intersection as the incentre.

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