1. **State the problem:** We have two triangles ABC and CDE sharing point C, with points A, C, D collinear and points B, C, E collinear. Given lengths AC = 24 cm, BC = 31 cm, CE = 19 cm, CD = 16 cm, and angle BAC = 64°, we need to find the length DE. 2. **Analyze the problem:** Since A, C, D are collinear and B, C, E are collinear, triangles ABC and CDE share line segments and angles. We can use the Law of Cosines and properties of triangles to find DE. 3. **Find length AB using Law of Cosines in triangle ABC:** $$AB^2 = AC^2 + BC^2 - 2 \times AC \times BC \times \cos(\angle BAC)$$ Substitute values: $$AB^2 = 24^2 + 31^2 - 2 \times 24 \times 31 \times \cos(64^\circ)$$ Calculate: $$AB^2 = 576 + 961 - 2 \times 24 \times 31 \times 0.4384$$ $$AB^2 = 1537 - 2 \times 24 \times 31 \times 0.4384$$ Calculate the product: $$2 \times 24 \times 31 = 1488$$ $$1488 \times 0.4384 = 652.5$$ So: $$AB^2 = 1537 - 652.5 = 884.5$$ Therefore: $$AB = \sqrt{884.5} \approx 29.74\text{ cm}$$ 4. **Find angle ABC in triangle ABC:** Use Law of Cosines again to find angle ABC: $$\cos(\angle ABC) = \frac{AB^2 + BC^2 - AC^2}{2 \times AB \times BC}$$ Substitute values: $$\cos(\angle ABC) = \frac{29.74^2 + 31^2 - 24^2}{2 \times 29.74 \times 31}$$ Calculate numerator: $$29.74^2 = 884.5, \quad 31^2 = 961, \quad 24^2 = 576$$ $$884.5 + 961 - 576 = 1269.5$$ Calculate denominator: $$2 \times 29.74 \times 31 = 1843.88$$ So: $$\cos(\angle ABC) = \frac{1269.5}{1843.88} = 0.6885$$ Therefore: $$\angle ABC = \cos^{-1}(0.6885) \approx 46.5^\circ$$ 5. **Find angle BCE in line B-C-E:** Since B, C, E are collinear, angle BCE is a straight line, so: $$\angle BCE = 180^\circ - \angle ABC = 180^\circ - 46.5^\circ = 133.5^\circ$$ 6. **Use Law of Cosines in triangle CDE to find DE:** Given CD = 16 cm, CE = 19 cm, and angle DCE = 133.5° (angle between CD and CE), apply Law of Cosines: $$DE^2 = CD^2 + CE^2 - 2 \times CD \times CE \times \cos(133.5^\circ)$$ Calculate: $$DE^2 = 16^2 + 19^2 - 2 \times 16 \times 19 \times \cos(133.5^\circ)$$ Calculate squares: $$16^2 = 256, \quad 19^2 = 361$$ Calculate cosine: $$\cos(133.5^\circ) = -0.676$$ Calculate product: $$2 \times 16 \times 19 = 608$$ So: $$DE^2 = 256 + 361 - 608 \times (-0.676) = 617 + 411.01 = 1028.01$$ Therefore: $$DE = \sqrt{1028.01} \approx 32.06\text{ cm}$$ 7. **Final answer:** Length of DE is approximately **32.1 cm** correct to 3 significant figures.