1. **State the problem:** We have a circle with a secant line FED and a tangent line CD. Given the lengths CD = 43 and ED = 31, we need to find the length of FE. 2. **Recall the tangent-secant theorem:** The square of the length of the tangent segment (CD) is equal to the product of the lengths of the entire secant segment (FED) and its external part (ED). Mathematically, this is: $$CD^2 = FE \times FED$$ Since FED is the entire secant segment, it can be expressed as: $$FED = FE + ED$$ 3. **Set up the equation:** Substitute the known values: $$43^2 = FE \times (FE + 31)$$ 4. **Simplify and form a quadratic equation:** $$1849 = FE^2 + 31FE$$ Rearranged: $$FE^2 + 31FE - 1849 = 0$$ 5. **Solve the quadratic equation using the quadratic formula:** $$FE = \frac{-31 \pm \sqrt{31^2 - 4 \times 1 \times (-1849)}}{2}$$ Calculate the discriminant: $$31^2 = 961$$ $$4 \times 1 \times 1849 = 7396$$ $$\sqrt{961 + 7396} = \sqrt{8357} \approx 91.43$$ 6. **Calculate the two possible solutions:** $$FE = \frac{-31 + 91.43}{2} = \frac{60.43}{2} = 30.215$$ $$FE = \frac{-31 - 91.43}{2} = \frac{-122.43}{2} = -61.215$$ Since length cannot be negative, we discard the negative solution. 7. **Final answer:** $$FE \approx 30.2$$ units (rounded to the nearest tenth). **Note:** The problem states the answer is 34.9 units, which suggests a possible difference in labeling or values. Based on the tangent-secant theorem and given lengths, the calculated length of FE is approximately 30.2 units.