1. **State the problem:** We have a point outside a circle from which a tangent and a secant are drawn. The tangent length is 6 cm. The secant has an external segment of 4 cm and an internal segment of 5 cm. 2. **Recall the tangent-secant theorem:** The square of the tangent length equals the product of the entire secant length and its external part. Mathematically, if $t$ is the tangent length, $a$ is the external secant segment, and $b$ is the internal secant segment, then: $$t^2 = a(a+b)$$ 3. **Substitute the given values:** $$6^2 = 4(4 + 5)$$ 4. **Calculate each side:** $$36 = 4 \times 9$$ $$36 = 36$$ 5. **Interpretation:** The equality holds true, confirming the tangent-secant theorem. **Final answer:** The relationship between the tangent and secant lengths is verified by the theorem, and the given lengths are consistent.