1. The problem involves creating an artistic design using circles by applying concepts of circle equations, centers, and radii. 2. The flower consists of one large center circle and four smaller petals represented by circles. 3. Writing the equation of each circle in standard form $(x - h)^2 + (y - k)^2 = r^2$: - Center of flower: $(0, 0)$, radius $5$ Equation: $$ (x - 0)^2 + (y - 0)^2 = 5^2 $$ Simplifies to $$ x^2 + y^2 = 25 $$ - Petal 1: center $(5, 0)$, radius $3$ Equation: $$ (x - 5)^2 + (y - 0)^2 = 3^2 $$ Simplifies to $$ (x - 5)^2 + y^2 = 9 $$ - Petal 2: center $(-5, 0)$, radius $3$ Equation: $$ (x + 5)^2 + (y - 0)^2 = 3^2 $$ Simplifies to $$ (x + 5)^2 + y^2 = 9 $$ - Petal 3: center $(0, 5)$, radius $3$ Equation: $$ (x - 0)^2 + (y - 5)^2 = 3^2 $$ Simplifies to $$ x^2 + (y - 5)^2 = 9 $$ - Petal 4: center $(0, -5)$, radius $3$ Equation: $$ (x - 0)^2 + (y + 5)^2 = 3^2 $$ Simplifies to $$ x^2 + (y + 5)^2 = 9 $$ 4. This configuration forms a symmetrical flower pattern. 5. The decorative floral border consists of multiple colorful flower shapes arranged horizontally with circular spirals and petals. 6. Reflection: I used symmetry by placing centers of petals equidistant from the origin to create a balanced flower shape. I used varying radii to distinguish the central flower and its petals. I applied the standard circle equation form to accurately define positions and sizes. Final answer includes equations: Center circle: $$x^2 + y^2 = 25$$ Petals: $$ (x - 5)^2 + y^2 = 9 $$, $$ (x + 5)^2 + y^2 = 9 $$, $$ x^2 + (y - 5)^2 = 9 $$, $$ x^2 + (y + 5)^2 = 9 $$ These equations describe the flower and petals precisely for graphing or artistic design.