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To find the length of \( HG \) in the given diagram, we will again use the Power of a Point theorem.

Given:
- \( EF \) is the tangent line.
- \( HGF \) is the secant line with \( HG \) as the internal segment and \( GF \) as the external segment.

According to the Power of a Point theorem:
\[ EF^2 = HG \cdot HF \]

From the diagram:
- \( EF = 11 \)
- \( GF = 7 \)

Let's denote \( HG \) as \( x \). The full length of the secant \( HF \) is the sum of \( HG \) and \( GF \):
\[ HF = x + 7 \]

Using the Power of a Point theorem:
\[ 11^2 = x \cdot (x + 7) \]
\[ 121 = x^2 + 7x \]

This is a quadratic equation. To solve it, we need to rearrange it:
\[ x^2 + 7x - 121 = 0 \]

We can solve this quadratic equation using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 1 \), \( b = 7 \), and \( c = -121 \).

Plugging in these values:
\[ x = \frac{-7 \pm \sqrt{7^2 - 4 \cdot 1 \cdot (-121)}}{2 \cdot 1} \]
\[ x = \frac{-7 \pm \sqrt{49 + 484}}{2} \]
\[ x = \frac{-7 \pm \sqrt{533}}{2} \]
\[ x = \frac{-7 \pm 23.1}{2} \]

Considering the positive root, we get:
\[ x = \frac{-7 + 23.1}{2} \]
\[ x = \frac{16.1}{2} \]
\[ x \approx 8.05 \]

Thus, the length of \( HG \) is approximately 8.1 units (rounded to the nearest tenth).

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: When solving quadratic equations, always check if the quadratic formula can be simplified or if there is an opportunity to factor the equation for a quicker solution.

Learn how to calculate the length of segment HG using the Power of a Point theorem with a detailed step-by-step solution. This problem involves circle geometry, quadratic equations, and the application of the Power of a Point theorem. Suitable for high school students (Grades 9-12).

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