This diagram shows two intersecting chords inside a circle. To find the value of $x$, we can apply the chord length theorem (also known as the intersecting chords theorem), which states:

$\text{(part 1 of chord 1)} \times \text{(part 2 of chord 1)} = \text{(part 1 of chord 2)} \times \text{(part 2 of chord 2)}$

From the diagram, we have:

• One chord is divided into lengths $2.7$ and $x$,
• The other chord is divided into lengths $4.9$ and $10.6$.

Using the theorem, the equation becomes:

$2.7 \times x = 4.9 \times 10.6$

Let's solve for $x$.The value of $x$ is approximately $19.24$ (rounded to two decimal places).

Would you like more details on this, or do you have any further questions?

Here are some related questions to consider:

1. How is the intersecting chords theorem derived?
2. What other geometric properties of circles involve chords and angles?
3. How would the solution change if the intersecting point of the chords were at a different location?
4. Can you apply the intersecting chords theorem to secants as well?
5. What other theorems can be used to find unknown values in circle-related problems?

Tip: When solving problems with circles, keep in mind that many theorems are based on relationships between angles, lengths, and radii.