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Given the circle below with secant start overline, Z, Y, X, end overline 
ZYX
  and tangent start overline, W, X, end overline 
WX
 , find the length of start overline, W, X, end overline 
WX
 . Round to the nearest tenth if necessary.


To find the length of \( WX \) in the given diagram, we can use the Power of a Point theorem, which states that for a point \( P \) outside a circle, the product of the lengths of the segments of any secant line passing through \( P \) is equal to the square of the length of the tangent segment from \( P \) to the circle.

In this problem, point \( X \) is outside the circle, and \( ZX \) is the secant line intersecting the circle at points \( Z \) and \( Y \). \( WX \) is the tangent line from \( X \) to the circle.

The Power of a Point theorem can be written as:
\[ WX^2 = XY \cdot XZ \]

From the diagram:
- \( XY = 15 \)
- \( XZ = XY + YZ = 15 + 29 = 44 \)

We can now substitute these values into the equation:
\[ WX^2 = 15 \cdot 44 \]
\[ WX^2 = 660 \]

Taking the square root of both sides to find \( WX \):
\[ WX = \sqrt{660} \]
\[ WX \approx 25.7 \]

So, the length of \( WX \) is approximately 25.7 units.

Would you like further details on any part of this solution or have any additional questions?

Here are some questions you might want to ask:
1. How do you derive the Power of a Point theorem?
2. Can you explain the steps for solving similar problems with different given values?
3. What are some other applications of the Power of a Point theorem?
4. How can this theorem be applied in three-dimensional geometry?
5. What are other important theorems related to circles?

Tip: Always double-check your calculations and ensure you understand the underlying principles behind the formulas you use.

Learn how to find the length of tangent WX in a circle using the Power of a Point theorem. Detailed step-by-step solution involving secant and tangent lines intersecting a circle, suitable for high school geometry students.

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