In this problem, we can use the Secant-Tangent Theorem, which states that for two secants (or two chords) intersecting outside the circle, the products of the lengths of the segments of the secants are equal.
The formula we will use is:
\[ SR \cdot SQ = ST \cdot SU \]
Given the lengths:
- \( SR = 12 \)
- \( QR = 9 \) (thus, \( SQ = SR + QR = 12 + 9 = 21 \))
- \( ST = 6 \)
- \( TU = 8x - 12 \) (thus, \( SU = ST + TU = 6 + (8x - 12) = 8x - 6 \))
Now, substituting these values into the equation:
\[ SR \cdot SQ = ST \cdot SU \] \[ 12 \cdot 21 = 6 \cdot (8x - 6) \]
Calculating the left side:
\[ 12 \cdot 21 = 252 \]
Now substituting in for \( SU \):
\[ 252 = 6 \cdot (8x - 6) \]
Expanding the right side:
\[ 252 = 48x - 36 \]
Now, add 36 to both sides:
\[ 252 + 36 = 48x \] \[ 288 = 48x \]
Now, divide both sides by 48:
\[ x = \frac{288}{48} = 6 \]
The value of \( x \) is 6.
Thus, the correct response is:
x is 6 units.