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Find EG if FG = 8, EH = x - 1, and EG = x + 1

This problem involves a geometric diagram where \( E \), \( F \), \( G \), and \( H \) are points. From the image, it appears that line segments \( EF \), \( EG \), and \( EH \) are related to each other, with \( G \) being the center of a circle, and \( F \) and \( H \) are points on the circumference.

### Known Information:
- \( FG = 8 \) (radius of the circle)
- \( EH = x - 1 \)
- \( EG = x + 1 \)

We can assume \( E \) lies outside the circle, and since \( G \) is the center, segments \( FG \) and \( GH \) are radii of the circle.

Using the **Power of a Point theorem**, which states that for a point outside a circle, the product of the segments of one secant (from the external point to two points on the circle) equals the product of the segments of another secant, we can write:

\[
EG \cdot EH = EF \cdot EG
\]

This gives us the following relation:

\[
(x + 1)(x - 1) = 8^2
\]

### Solving the equation:

\[
(x + 1)(x - 1) = 64
\]

This simplifies to:

\[
x^2 - 1 = 64
\]

Now, add 1 to both sides:

\[
x^2 = 65
\]

Finally, take the square root of both sides:

\[
x = \sqrt{65}
\]

Therefore, the value of \( x \) is \( \sqrt{65} \).

### Finding \( EG \):
Since \( EG = x + 1 \), substitute \( x \):

\[
EG = \sqrt{65} + 1
\]

So, \( EG = \sqrt{65} + 1 \).

Would you like further explanation or details?

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Here are 5 related questions to deepen understanding:
1. How do you apply the Power of a Point theorem in different circle configurations?
2. What are the different types of circle theorems used to solve geometry problems?
3. How can you approximate the value of \( \sqrt{65} \)?
4. What happens if the external point is tangential to the circle instead of forming secants?
5. How do we find the length of the other segments in the circle based on given lengths?

**Tip:** Remember that the Power of a Point theorem applies only when the point lies outside the circle.

Find EG if FG = 8, EH = x - 1, and EG = x + 1.

Find the length of segment EG in a geometry problem involving a circle and external point. Given FG = 8, EH = x - 1, and EG = x + 1, we solve using the Power of a Point Theorem. Suitable for students in Grades 9-11.

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