1. **Problem statement:** We have a right-angled triangle XYZ with the right angle at Y. Side XY measures 5 cm, side XZ (the hypotenuse) measures 19 cm, and we need to find the length of side YZ. 2. **Formula used:** According to Pythagoras' theorem, in a right-angled triangle, $$XZ^2 = XY^2 + YZ^2$$ where $XZ$ is the hypotenuse, and $XY$ and $YZ$ are the other two sides. 3. **Substitute known values:** $$19^2 = 5^2 + YZ^2$$ 4. **Calculate squares:** $$361 = 25 + YZ^2$$ 5. **Isolate $YZ^2$:** $$YZ^2 = 361 - 25$$ $$YZ^2 = 336$$ 6. **Find $YZ$ by taking the square root:** $$YZ = \sqrt{336}$$ 7. **Simplify the square root:** $$YZ = \sqrt{16 \times 21} = 4\sqrt{21}$$ 8. **Calculate decimal value:** $$YZ \approx 4 \times 4.5826 = 18.3304$$ 9. **Round to 1 decimal place:** $$YZ \approx 18.3\text{ cm}$$ **Final answer:** The length of side YZ is approximately **18.3 cm**.