ABC is an equilarteral triangle of side 6 cm. If a circle of radius 1 cm is moving inside and along the sides of the triangle, then locus of the centre of the circle is an equilateral triangle of side _________.

Options

A. 5 cm
B. 4cm
C. `(6-2sqrt(3))cm`
D. `(3+sqrt(3))cm`

Explanation

To find the side length of the equilateral triangle formed by the locus of the center of a circle of radius 1 cm moving inside an equilateral triangle of side 6 cm, we can follow these steps:

### Step 1: Understand the Problem
We have an equilateral triangle ABC with a side length of 6 cm. A circle with a radius of 1 cm is moving inside this triangle. The center of the circle will trace out a smaller equilateral triangle as it moves.

**Hint:** Visualize the original triangle and the circle moving inside it.

### Step 2: Determine the Distance from the Sides
The center of the circle will be at a distance of 1 cm from each side of the triangle since the radius of the circle is 1 cm. Therefore, the locus of the center will be an equilateral triangle that is smaller than triangle ABC.

**Hint:** Remember that the center of the circle must always be 1 cm away from the sides of triangle ABC.

### Step 3: Calculate the New Side Length
To find the side length of the new triangle formed by the locus of the center, we need to subtract twice the radius of the circle from the original side length of triangle ABC. This is because the circle's center moves 1 cm inward from each side.

The formula to find the side length of the new triangle is:
\[ \text{New Side Length} = \text{Original Side Length} - 2 \times \text{Radius} \]

Substituting the values:
\[ \text{New Side Length} = 6 \, \text{cm} - 2 \times 1 \, \text{cm} \]
\[ \text{New Side Length} = 6 \, \text{cm} - 2 \, \text{cm} \]
\[ \text{New Side Length} = 4 \, \text{cm} \]

**Hint:** Make sure to subtract the radius from both sides of the triangle.

### Step 4: Conclusion
The side length of the equilateral triangle formed by the locus of the center of the circle is 4 cm.

**Final Answer:** The side of the equilateral triangle is **4 cm**.