The problem provides an equilateral triangle with three circles of radius $1$ cm each, placed inside. The vertices of the triangle have coordinates $(2, 4.732)$, $(-0.732, 0)$, and $(4.732, 0)$.

Steps to Approach the Problem:

1. Understanding Triangle Geometry: Since it’s an equilateral triangle, each side is equal. The coordinates of the vertices can help us calculate the side length by using the distance formula between any two points. This information may help us verify the setup for the circles.

2. Circle Placement: The circles are arranged such that they are tangent to each other and to the sides of the triangle. Given the radius of each circle as $1$ cm, we can conclude:

   • The centers of the circles must form an equilateral triangle within the main triangle.
   • The side length of this inner equilateral triangle (formed by the centers of the circles) should be $2$ cm because each side is the sum of the radii of two adjacent circles.

3. Calculating the Side Length of the Triangle: Using the coordinates, let’s calculate the side length:

   • Between $(-0.732, 0)$ and $(4.732, 0)$: $\text{Distance} = \sqrt{(4.732 - (-0.732))^2 + (0 - 0)^2} = \sqrt{5.464^2} = 5.464 \text{ cm}$ This is the length of each side of the equilateral triangle.

4. Height Verification: The height $h$ of an equilateral triangle with side length $a$ is given by: $h = \frac{\sqrt{3}}{2} a$ Substituting $a = 5.464$: $h = \frac{\sqrt{3}}{2} \times 5.464 \approx 4.732 \text{ cm}$ This matches the y-coordinate of the top vertex, confirming the triangle’s dimensions align with the given coordinates.

5. Circle Placement Verification: With a side length of $5.464$ cm for the outer triangle, the three circles of radius $1$ cm each fit within, touching each other and the triangle sides.

Would you like more details on any part of this solution?

Here are some related questions to deepen understanding:

1. How can we calculate the area of this equilateral triangle?
2. What would the area of each circle be, and how does it compare to the triangle's area?
3. If we inscribe a fourth circle that touches the three inner circles, what would its radius be?
4. How can we derive the coordinates of the centers of each circle within the triangle?
5. What are the angles formed between the lines connecting the triangle's vertices and the centers of the circles?

Tip: In equilateral triangles, the relationship between side length and height is consistent, allowing for easy verification of dimensions using the ratio $\sqrt{3}$.