To solve for $DC$ in this problem, we can use the intersecting chords theorem. The theorem states that if two chords intersect inside a circle, the products of the lengths of the segments of each chord are equal.

In this problem:

• The two intersecting chords are $AB$ and $CD$.
• Given: $AB = 16 \, \text{cm}, \quad BP = 14 \, \text{cm}, \quad CP = 12 \, \text{cm}$

Let’s denote:

• $AP = AB + BP = 16 + 14 = 30 \, \text{cm}$
• Let $DP = DC + CP$, where $DC$ is what we want to find.

According to the intersecting chords theorem: $AB \times BP = CP \times DC$ Substituting the given values: $16 \times 14 = 12 \times DC$ $224 = 12 \times DC$ $DC = \frac{224}{12} = 18.67 \, \text{cm}$

Therefore, the length of $DC$ is approximately $18.67 \, \text{cm}$.

Would you like further clarification, or do you have any questions?

Here are some related questions:

1. How does the intersecting chords theorem apply to other shapes?
2. What are some alternative methods to find unknown lengths in circle geometry?
3. How can the intersecting chords theorem be proven?
4. Are there situations where the intersecting chords theorem does not apply?
5. Can this theorem be extended to non-circular shapes?

Tip: The intersecting chords theorem is useful for any configuration where two chords intersect, whether or not they form straight lines outside the circle.