Chapter 10.5, Problem 15ES

Let an $x\text{'}y\text{'}\text{-coordinate}$ system be obtained by rotating an $xy\text{-coordinate}$ system through an angle $\theta .$ Prove: For every value of $\theta ,$ the equation $x^2+y^2=r^2$ becomes the equation $x{\text{'}}^2+y{\text{'}}^2=r^2.$ Give a geometric explanation.