TechnoWizard

Let's try setting x equal to the length of each leg. Then, we have two sides of length x and one of length \( \dfrac 32\cdot x,\) so the perimeter is \(x+x+\dfrac 32\cdot x, \)which is  \(\dfrac 72\cdot x.\)We also know the perimeter is  63 so we set our two expressions for the perimeter equal to each other: \(\frac 72\cdot x = 63.\)
Multiplying both sides by \(\frac{2}{7}\) gives  The base is \(1\frac 12\cdot 18 = \boxed{27}\)
 

Hope it helps! 

\(0.1\overline{23}\) is the same thing as 122/990. Since they have to have a GCD of 1, we must simplify them, where we get 61/495. Now we add the numerator and denominator getting \(61+495 = \boxed{556}\)

I know that this question was asked over half a year ago, but hope it helps!