Question

Twelve synchronized swimmers are forming a circle. The location of those swimmers are (13,-2) (-1,-2) and (6,-9). A 4th swimmer will appear in the center of the circle. Where would the center swimmers need to be located?

Answer 1

Let the location of the center (x,y)
let the distance between center and point (13,-2) = d₁
let the distance between center and point (-1,-2) = d₂
let the distance between center and point (6,-9) = d₃

The condition of the circle is d₁ = d₂ = d₃
Making two equations to find x , y

d₁² = d₂²
∴ (x-13)² + (y+2)² = (x+1)² + (y+2)²
∴ (x-13)²  = (x+1)²
∴ x-13 = $\pm$ (x+1) ⇒⇒⇒ positive sign unacceptable
∴ x-13 = -(x+1)
∴ x = 6

d₁² = d₃²
∴ (x-13)² + (y+2)² = (x-6)² + (y+9)²
substituting with x and simplifying the equation
∴ y = -2

∴center = (6,-2)

The center swimmers need to be located at (6,-2)