R+r = \frac{c}{2} + \frac{ab}{a+b+c} = \frac{c(a+b+c)+2ab}{2(a+b+c)} = \frac{10c+c^2+2ab}{2(10+c)} = \frac{10c+a^2+b^2+2ab}{2(c+10)} = \frac{10c+(a+b)^2}{2(c+10)} = \frac{10(c+10)}{2(c+10)} = 5