Complete the equation for the circle with center \( (-2,-4) \) and radius 9 . \( (x+2 \)

The equation of a circle can be formulated with the center coordinates and the radius. For a circle with center \( (-2, -4) \) and radius 9, the equation is: \[ (x + 2)^2 + (y + 4)^2 = 81. \] So, you just need to complete it by adding \( (y + 4)^2 \) and setting it equal to \( 9^2 = 81 \). Did you know that circles are not just beautiful geometrical shapes but have been studied for centuries? The ancient Greeks, especially mathematicians like Euclid and Archimedes, made significant strides in understanding their properties and applications. Their studies laid the foundational work for geometry, which still plays a crucial role in modern mathematics! If you're trying to visualize this equation, it's a great idea to graph it! Start by plotting the center at \( (-2, -4) \) on a coordinate plane. Then, measure a distance of 9 units in all directions (up, down, left, right). Connecting those points will form your perfect circle! Just remember that each point on the circle maintains the same distance from the center—it's all about that radius!