Question

the equation ( a=\frac{180(n - 2)}{n} ) represents the angle measures, ( a ), in a regular ( n ) - sided polygon. when the equation is solved for ( n ), ( n ) is equal to a fraction with a denominator of ( a - 180 ). what is the numerator of the fraction?

Explanation:

Step1: Cross - multiply the equation

Given \(a=\frac{180(n - 2)}{n}\), cross - multiply to get \(an=180(n - 2)\).

Step2: Expand the right - hand side

Expand \(180(n - 2)\) to \(an = 180n-360\).

Step3: Group the terms with \(n\)

Move the terms with \(n\) to one side: \(an-180n=- 360\).

Step4: Factor out \(n\)

Factor out \(n\) from the left - hand side: \(n(a - 180)=-360\).

Step5: Solve for \(n\)

Divide both sides by \((a - 180)\) to get \(n=\frac{-360}{a - 180}\).

Answer:

\(-360\)