To solve the problem shown in the image involving segment lengths within a circle, we use the Power of a Point theorem. This theorem states:

If a tangent segment and a secant segment are drawn to a circle from the same external point, then:

$\text{(Tangent Segment)}^2 = \text{(External Segment)} \times \text{(Total Secant Length)}.$

Given:

• Tangent segment = $x$,
• External segment of the secant = $4$,
• Total secant length = $4 + 8 = 12$.

Applying the formula:

$x^2 = 4 \cdot 12.$

Simplify: $x^2 = 48.$

Take the square root: $x = \sqrt{48}.$

Simplify further: $x = 4\sqrt{3}.$

Final Answer:

$x = 4\sqrt{3} \approx 6.93 \, \text{(rounded to two decimal places)}.$

Would you like further explanation or details about the steps? Here are five related questions for exploration:

1. How is the Power of a Point theorem applied in other cases, such as intersecting chords?
2. What are the differences between tangent, secant, and chord properties in circles?
3. Can this theorem be extended to solve real-world problems involving circles?
4. How can one derive the Power of a Point theorem geometrically?
5. What are practical tips to simplify square roots in geometric problems?

Tip: Always check whether the external segment and the total length of the secant are clearly defined to avoid calculation errors!