This problem involves two segments that intersect either inside or outside a circle. The formula provided states that if two segments intersect, the product of the lengths of the segments on one side equals the product of the lengths of the segments on the other side.

The segments are described as follows:

• The purple segment intersects the circle at a point and has parts of 2 cm and 6 cm.
• The blue segment intersects the circle and has parts of 3 cm and $d$ (the unknown length).

The relationship can be set up using the intersecting chords theorem: $(2 \, \text{cm}) \times (6 \, \text{cm}) = (3 \, \text{cm}) \times d$ Simplifying: $12 = 3d$ Now, solve for $d$: $d = \frac{12}{3} = 4 \, \text{cm}$

Thus, the length of segment $d$ is 4 cm.

Would you like further clarification or help with a related concept? Here are some additional questions:

1. How can the intersecting chords theorem be applied to problems with more than two intersecting segments?
2. What is the difference between this theorem and the power of a point theorem?
3. How would the result change if one of the segment lengths were different?
4. How can this formula be adapted for segments that intersect inside the circle?
5. Can this method be used in geometry problems that do not involve circles?

Tip: Always check for common geometric theorems, like the intersecting chords or power of a point theorem, when you encounter intersections in circle problems!