1. **Problem Statement:** We have a circle with two secants intersecting outside the circle at point H. One secant passes through points S and H, extending beyond H to F, with segment lengths $SH = 16$ and $HF = x$. The other secant passes through points T and F inside the circle, with segments $TF = 10$ and $F$ to the circle intersection point $= 8$. We need to find the length $x$ of segment $HF$. 2. **Relevant Formula:** For two secants intersecting outside a circle, the product of the entire length of one secant and its external segment equals the product of the entire length of the other secant and its external segment. Mathematically: $$SH \times HF = TF \times (TF + 8)$$ Here, $SH = 16$, $HF = x$, $TF = 10$, and the other segment on the second secant inside the circle is 8. 3. **Apply the formula:** $$16 \times x = 10 \times (10 + 8)$$ Simplify the right side: $$16x = 10 \times 18$$ $$16x = 180$$ 4. **Solve for $x$:** $$x = \frac{180}{16} = 11.25$$ 5. **Answer:** The length of segment $HF$ is $11.25$ units. This uses the property of secants intersecting outside a circle, where the product of the whole secant length and its external segment is constant for both secants.