1. **Problem statement:** Find the coordinates of the circumcenter of triangle J(5,0), K(5,-8), and L(0,0). Also find the equations of two perpendicular bisectors and their intersection point. 2. **Recall:** The circumcenter is the intersection of the perpendicular bisectors of the sides of a triangle. 3. **Step 1: Find midpoints of two sides.** - Midpoint of JK: $$\left(\frac{5+5}{2}, \frac{0+(-8)}{2}\right) = (5, -4)$$ - Midpoint of JL: $$\left(\frac{5+0}{2}, \frac{0+0}{2}\right) = (2.5, 0)$$ 4. **Step 2: Find slopes of sides JK and JL.** - Slope of JK: $$\frac{-8-0}{5-5} = \frac{-8}{0} = \text{undefined (vertical line)}$$ - Slope of JL: $$\frac{0-0}{0-5} = 0$$ 5. **Step 3: Find slopes of perpendicular bisectors.** - Perpendicular bisector of JK: Since JK is vertical, its perpendicular bisector is horizontal with slope 0. - Perpendicular bisector of JL: Since JL has slope 0, its perpendicular bisector is vertical with undefined slope. 6. **Step 4: Write equations of perpendicular bisectors.** - Perpendicular bisector of JK passes through midpoint (5, -4) with slope 0: $$y = -4$$ - Perpendicular bisector of JL passes through midpoint (2.5, 0) and is vertical: $$x = 2.5$$ 7. **Step 5: Find intersection of perpendicular bisectors.** - Solve system: $$y = -4$$ $$x = 2.5$$ - Intersection point is $$ (2.5, -4) $$ 8. **Step 6: Conclusion:** - The circumcenter is located at $$ (2.5, -4) $$. **Final answers:** - Equation of one perpendicular bisector: $$y = -4$$ - Equation of another perpendicular bisector: $$x = 2.5$$ - Intersection point: $$(2.5, -4)$$ - Circumcenter coordinates: $$(2.5, -4)$$