1. **State the problem:** We are given a triangle with sides |CF| = 22 cm, |EF| = 11.8 cm, and the angle $\angle EFC = 110^\circ$. We need to find the distance |CE| using the cosine rule. 2. **Recall the cosine rule:** For any triangle with sides $a$, $b$, and $c$, and angle $\theta$ opposite side $c$, the cosine rule states: $$c^2 = a^2 + b^2 - 2ab \cos(\theta)$$ 3. **Identify the sides and angle:** Here, $|CE|$ is the side opposite the angle $\angle EFC = 110^\circ$. The other two sides are $|CF| = 22$ cm and $|EF| = 11.8$ cm. 4. **Apply the cosine rule:** $$|CE|^2 = |CF|^2 + |EF|^2 - 2 \times |CF| \times |EF| \times \cos(110^\circ)$$ Substitute the values: $$|CE|^2 = 22^2 + 11.8^2 - 2 \times 22 \times 11.8 \times \cos(110^\circ)$$ 5. **Calculate each term:** $$22^2 = 484$$ $$11.8^2 = 139.24$$ 6. **Calculate the cosine term:** $$\cos(110^\circ) = \cos(110 \times \frac{\pi}{180}) \approx -0.3420$$ 7. **Calculate the product:** $$2 \times 22 \times 11.8 \times (-0.3420) = 2 \times 22 \times 11.8 \times -0.3420$$ $$= 519.2 \times -0.3420 = -177.5$$ 8. **Substitute back:** $$|CE|^2 = 484 + 139.24 - (-177.5) = 484 + 139.24 + 177.5 = 800.74$$ 9. **Find |CE|:** $$|CE| = \sqrt{800.74} \approx 28.3 \text{ cm}$$ **Final answer:** $$|CE| \approx 28.3 \text{ cm}$$