To solve problem 18 from the image, let's analyze the geometry and algebra involved. The problem involves a tangent-segment relation.

Problem:

You need to find $DG$, where $DG = 2x - 4$ and $GF = x + 7$. Since the segments $DG$ and $GF$ are tangents from the same external point $G$ to the circle, their lengths are equal. This leads to the equation:

$DG = GF$

Steps to Solve:

1. Substitute the given expressions for $DG$ and $GF$: $2x - 4 = x + 7$

2. Simplify the equation to isolate $x$: $2x - x = 7 + 4$ $x = 11$

3. Use $x = 11$ to find $DG$: $DG = 2x - 4 = 2(11) - 4 = 22 - 4 = 18$

Final Answer:

$DG = 18$

Would you like a detailed explanation of why tangent segments from an external point are equal or help with another problem?

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Here are 5 related questions to deepen your understanding:

1. Why are tangent segments from the same external point always equal?
2. How would the equation change if $DG = 3x + 1$ and $GF = x + 7$?
3. Can you solve for $x$ if $DG = 4x - 5$ and $GF = 2x + 9$?
4. What are the real-world applications of tangent properties in geometry?
5. How would you calculate the perimeter of a triangle formed by two tangent segments and a chord?

Tip:

When solving tangent problems, always verify the equality of segments drawn from the same external point!