To solve the problem, we can use the secant segments theorem, which states that if two secants intersect a circle, the product of the lengths of the whole secant and its external segment are equal.
In this case, we have two secants:
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For secant SQ:
- The whole length \( SQ \) is 30.
- The external segment \( SR \) is 12.
The length of the internal segment \( ST \) can be found as follows: \[ ST = SQ - SR = 30 - 12 = 18 \]
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For secant SU:
- The external segment \( SU \) is \( 5x - 24 \).
- The internal segment \( ST \) is 9.
Using the secant segments theorem, we can set up the equation: \[ SR \cdot ST = TU \cdot SU \] Substituting the known values: \[ 12 \cdot 9 = (5x - 24) \cdot 5 \]
This simplifies to: \[ 108 = (5x - 24) \cdot 5 \] Divide both sides by 5: \[ 21.6 = 5x - 24 \] Now, solving for \( x \): \[ 5x = 21.6 + 24 \] \[ 5x = 45.6 \] \[ x = \frac{45.6}{5} \] \[ x = 9.12 \]
This result does not match any of the provided answers. It appears I made an error in interpreting the relationships or calculating the segments. Let's verify.
Using \( TU = 5x - 24 \) as it stands and the relationships we have established correctly.
Substituting accurately for the secants and solving accurately leads to direct equations that match as established.
From checks we simplified:
- \( 12 \cdot 9 = (5x - 24) \cdot 5 \)
- Verify calculations on both ends yield values.
To finalize with x:
No provided answers align with direct calculation. Given results for consideration within an exact method approach can finalize using:
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