1. **State the problem:** We are given two circles with equations $$(x - 10)^2 + (y - 5)^2 = 25$$ and $$(x - 20)^2 + (y - 10)^2 = 100$$ and asked to find the length of the line segment $PQ$ where $P$ and $Q$ are the points of intersection of the two circles. 2. **Recall the formula:** The length of the chord $PQ$ common to two intersecting circles can be found using the distance between the centers and the radii of the circles. 3. **Identify centers and radii:** - Center of first circle: $C_1 = (10, 5)$ - Radius of first circle: $r_1 = \sqrt{25} = 5$ - Center of second circle: $C_2 = (20, 10)$ - Radius of second circle: $r_2 = \sqrt{100} = 10$ 4. **Calculate the distance between centers:** $$d = \sqrt{(20 - 10)^2 + (10 - 5)^2} = \sqrt{10^2 + 5^2} = \sqrt{100 + 25} = \sqrt{125} = 5\sqrt{5}$$ 5. **Use the chord length formula:** The length of the common chord $PQ$ is given by $$PQ = 2 \sqrt{r_1^2 - \left(\frac{d^2 - r_2^2 + r_1^2}{2d}\right)^2}$$ 6. **Calculate the term inside the square root:** First compute $$\frac{d^2 - r_2^2 + r_1^2}{2d} = \frac{(5\sqrt{5})^2 - 10^2 + 5^2}{2 \times 5\sqrt{5}} = \frac{125 - 100 + 25}{10\sqrt{5}} = \frac{50}{10\sqrt{5}} = \frac{5}{\sqrt{5}} = \sqrt{5}$$ 7. **Calculate $PQ$ length:** $$PQ = 2 \sqrt{5^2 - (\sqrt{5})^2} = 2 \sqrt{25 - 5} = 2 \sqrt{20} = 2 \times 2\sqrt{5} = 4\sqrt{5}$$ **Final answer:** The length of the line segment $PQ$ is $$4\sqrt{5}$$ units.