The image shows a geometric configuration with a circle and secants. To find the value of $x$, we can apply the Secant-Secant Power Theorem.

Theorem Explanation:

For two secants intersecting outside of a circle, the product of the entire length of one secant and its external part equals the product of the entire length of the other secant and its external part.

The secants in the diagram seem to be:

• Secant 1: $(x + 30)$
• Secant 2: $(x + 10)$ Both of these secants extend outside of the circle.

Using the Secant-Secant Power Theorem:

Let the point of intersection of the two secants outside the circle be $P$, and the points where the secants intersect the circle be $A, B, C,$ and $D$. Applying the theorem: $PA \times PB = PC \times PD$ Given the lengths from the diagram, we substitute: $(x + 30) \times x = (x + 10) \times (x + 10)$

Now, let's solve this equation step by step.

$(x + 30) \times x = (x + 10)^2$

Expanding both sides:

$x(x + 30) = (x + 10)(x + 10)$ $x^2 + 30x = x^2 + 20x + 100$

Simplifying:

$x^2 + 30x = x^2 + 20x + 100$ Subtract $x^2$ from both sides: $30x = 20x + 100$ Subtract $20x$ from both sides: $10x = 100$ Now, divide by 10: $x = 10$

Conclusion:

The value of $x$ is $\boxed{10}$.

Would you like further details on this process, or have any additional questions?

Here are five related questions to extend your understanding:

1. How does the Secant-Tangent Theorem differ from the Secant-Secant Theorem?
2. What other methods can be used to solve problems involving circles and secants?
3. How would the calculation change if one secant was a tangent?
4. What is the significance of external segments in circle theorems?
5. Can this method be applied to ellipses or other conic sections?

Tip: When solving geometric problems involving circles, always verify if you can apply one of the key theorems like the Secant-Secant, Tangent-Secant, or Inscribed Angle Theorem.