Solve (R+r):2R=1:sqrt{2}

$$( R + r ) : 2 R = 1 : \sqrt { 2 }$$

Solve for r

$$r=-R+\frac{\sqrt{2}}{R}$$

$$R\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2.

$$\left(R+r\right)R=2^{\frac{1}{2}}$$

Use the distributive property to multiply R+r by R.

$$R^{2}+rR=2^{\frac{1}{2}}$$

Subtract R^{2} from both sides.

$$rR=2^{\frac{1}{2}}-R^{2}$$

Reorder the terms.

$$Rr=-R^{2}+\sqrt{2}$$

Divide both sides by R.

$$\frac{Rr}{R}=\frac{-R^{2}+\sqrt{2}}{R}$$

Dividing by R undoes the multiplication by R.

$$r=\frac{-R^{2}+\sqrt{2}}{R}$$

Divide -R^{2}+\sqrt{2} by R.

$$r=-R+\frac{\sqrt{2}}{R}$$

Solve for R

$$R=\frac{\sqrt{r^{2}+4\sqrt{2}}-r}{2}$$

$$R=\frac{-\sqrt{r^{2}+4\sqrt{2}}-r}{2}$$