To solve the problem involving the chords $XY$ and $PQ$ intersecting externally, we can use the property of intersecting chords:

If two chords $XY$ and $PQ$ intersect outside the circle, the following relationship holds:

$XY \times YZ = PQ \times QZ$

Here, we know:

• $XY = 5 \, \text{cm}$
• $PQ = 3 \, \text{cm}$
• $QZ = 4 \, \text{cm}$

Let $ZY = x$ (the length we want to find). Therefore, we can rewrite the relationship as:

$XY \times ZY = PQ \times QZ$

Substituting the known values:

$5 \times x = 3 \times 4$

Calculating the right side:

$5x = 12$

Now, solving for $x$:

$x = \frac{12}{5} = 2.4 \, \text{cm}$

Thus, the length of $ZY$ is $2.4 \, \text{cm}$.

If you have any questions or need further details, feel free to ask! Here are some related questions for deeper exploration:

1. What are other properties of intersecting chords in circles?
2. How does the length of a chord relate to the radius of a circle?
3. Can you provide examples of similar problems with different values?
4. What is the formula for the area of a triangle formed by two chords in a circle?
5. How would the solution change if the intersection was internal instead of external?

Tip: Always sketch the problem when working with geometry, as it helps visualize relationships between different elements!