1. **Problem Statement:** We have a secant line intersecting a circle at two points, creating segments outside and inside the circle. The segments outside the circle are labeled 9 and 3, and the chord inside the circle is labeled 12. We need to solve for the unknown segment length $x$ on the secant. 2. **Relevant Formula:** For a secant line intersecting a circle, the product of the whole secant segment and its external segment on one side equals the product of the whole secant segment and its external segment on the other side. This is expressed as: $$ (x + 12) \times x = 9 \times (9 + 12 + 3) $$ 3. **Explanation:** - The segment inside the circle is 12. - The external segments are 9 and 3. - The total length of the secant on one side is $x + 12$. - The total length of the secant on the other side is $9 + 12 + 3 = 24$. 4. **Set up the equation:** $$ (x + 12) \times x = 9 \times 24 $$ 5. **Simplify the right side:** $$ 9 \times 24 = 216 $$ 6. **Rewrite the equation:** $$ x(x + 12) = 216 $$ 7. **Expand the left side:** $$ x^2 + 12x = 216 $$ 8. **Bring all terms to one side:** $$ x^2 + 12x - 216 = 0 $$ 9. **Solve the quadratic equation using the quadratic formula:** $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ where $a=1$, $b=12$, and $c=-216$. 10. **Calculate the discriminant:** $$ \Delta = 12^2 - 4 \times 1 \times (-216) = 144 + 864 = 1008 $$ 11. **Calculate the square root:** $$ \sqrt{1008} = \sqrt{16 \times 63} = 4\sqrt{63} $$ 12. **Find the roots:** $$ x = \frac{-12 \pm 4\sqrt{63}}{2} = -6 \pm 2\sqrt{63} $$ 13. **Evaluate approximate values:** $$ 2\sqrt{63} \approx 2 \times 7.937 = 15.874 $$ 14. **Possible solutions:** $$ x_1 = -6 + 15.874 = 9.874 $$ $$ x_2 = -6 - 15.874 = -21.874 $$ 15. **Interpretation:** Since length cannot be negative, the solution is: $$ \boxed{9.874} $$ This is the length of the segment $x$ on the secant outside the circle.