1. **State the problem:** We are given a triangle $\Delta XYZ$ with side lengths $x = 850$ cm, $y = 460$ cm, and $z = 590$ cm. We need to find the area of this triangle to the nearest tenth of a square centimeter. 2. **Formula used:** To find the area of a triangle when all three sides are known, we use Heron's formula: $$\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$$ where $a, b, c$ are the side lengths and $s$ is the semi-perimeter: $$s = \frac{a + b + c}{2}$$ 3. **Calculate the semi-perimeter $s$:** $$s = \frac{850 + 460 + 590}{2} = \frac{1900}{2} = 950$$ 4. **Apply Heron's formula:** $$\text{Area} = \sqrt{950(950 - 850)(950 - 460)(950 - 590)}$$ Calculate each term inside the square root: $$950 - 850 = 100$$ $$950 - 460 = 490$$ $$950 - 590 = 360$$ So, $$\text{Area} = \sqrt{950 \times 100 \times 490 \times 360}$$ 5. **Simplify the product inside the square root:** $$950 \times 100 = 95000$$ $$490 \times 360 = 176400$$ So, $$\text{Area} = \sqrt{95000 \times 176400} = \sqrt{16758000000}$$ 6. **Calculate the square root:** $$\text{Area} \approx 129441.9$$ 7. **Final answer:** The area of $\Delta XYZ$ is approximately $129441.9$ square centimeters to the nearest tenth.