1. **State the problem:** We have a circle centered at point $O$ with tangent segments $DE$ and $DF$ from point $D$ to the circle. Given $OE = 8.1$ and $OD = 13.5$, we need to find the length of $DF$. 2. **Recall the tangent-segment theorem:** Tangents drawn from an external point to a circle are equal in length. Therefore, $DE = DF$. 3. **Analyze the given lengths:** $OE$ is the radius of the circle to the tangent point $E$, so the radius $r = OE = 8.1$. 4. **Use the right triangle formed:** Since $DE$ is tangent at $E$, $OE$ is perpendicular to $DE$. Triangle $ODE$ is right-angled at $E$. 5. **Apply the Pythagorean theorem:** In triangle $ODE$, with right angle at $E$, we have $$OD^2 = OE^2 + DE^2$$ Substitute the known values: $$13.5^2 = 8.1^2 + DE^2$$ 6. **Calculate $DE^2$:** $$DE^2 = 13.5^2 - 8.1^2 = 182.25 - 65.61 = 116.64$$ 7. **Find $DE$:** $$DE = \sqrt{116.64} = 10.8$$ 8. **Find $DF$:** Since $DE = DF$, we have $$DF = 10.8$$ **Final answer:** $$\boxed{10.8}$$