If the center of a circle is at \( (-7,9) \) and its radius is 5 , complete its equation: \( (x-[?])^{2}+(y-[\quad])^{2}=[ \)

To write the equation of a circle, you can use the standard form, which is \( (x - h)^{2} + (y - k)^{2} = r^{2} \), where \( (h,k) \) is the center and \( r \) is the radius. In this case, the center is \( (-7, 9) \), so \( h = -7 \) and \( k = 9 \). The radius \( r \) is 5, so \( r^{2} = 5^{2} = 25 \). Putting it all together, the completed equation of the circle is: \( (x + 7)^{2} + (y - 9)^{2} = 25 \). For math lovers, did you know that circles are not only fun shapes but also deeply rooted in ancient history? The ancient Greeks studied circles extensively, and mathematicians like Archimedes were instrumental in determining properties related to them. Did you know that the famous equation \( C = 2\pi r \) for the circumference of a circle was derived during these explorations? On a practical note, circles are everywhere! Whether you're designing logos, laying out gardens, or even creating wheel designs, the concept of circles helps ensure everything is symmetrical and visually appealing. Plus, understanding circle equations can help you with everything from plotting graphs to calculating areas in geometry – making it an essential skill for students and professionals alike!