Quadratic Equations| Algebra| ENGG10 LE3| Problem 1

If a regular polygon of ten sides is inscribed in a circle of radius r units, then, if s units is the length of the side,

\frac{r}{s}=\frac{s}{r-s}

Solve this formula for s in terms of r.

SOLUTION:

Multiply both sides by the LCD s\left(r-s\right)

r\left(r-s\right)=s\left(s\right)

r^2-rs=s^2

s^2+rs-r^2=0

Since we are solving for s in terms of r, we treat r as a constant. Thus, this is quadratic in s with the following:  a=1,\:b=r,\:c=-r^2

Solve for s using the quadratic formula

s=\frac{-b\pm \sqrt{b^2-4ac}}{2a}

s=\frac{-r\pm \sqrt{r^2-4\left(1\right)\left(-r^2\right)}}{2\left(1\right)}

s=\frac{-r\pm \sqrt{r^2+4r^2}}{2}

s=\frac{-r\pm \:\sqrt{5r^2}}{2}

s=\frac{-r\pm r\sqrt{5}}{2}

s=\frac{r\left(-1\pm \sqrt{5}\right)}{2}