Given:?A is the largest angle and ?B is the smallest angle.?A = 2?BThe length of the sides of the triangle are consecutive positive integers: x - 1, x, x + 1Formula used:The length of the side of the required triangle directly depends on the value of the angle opposite to that particular side, i.e., the longest side of the triangle is opposite to the largest angle and vice-versa.Sine rule:\(\frac{sinA}{a} = \frac{sinB}{b}= \frac{sinC}{c}\)Cosine rule:cosB = \(\frac{a^2 + c^2 - b^2}{2ac}\)Calculation:The sides correspond to the angles ?A, ?B and ?C of ?ABC are a, b and c respectively.According to the question,?A > ?C > ?B? a > c > bSo, a = x + 1c = xb = x - 1Now, by using the sine rule.? \(\frac{sin 2B}{x+1} = \frac{sin B}{x-1} \)? \(\frac{sin 2B}{x+1} = \frac{sin B}{x-1} \)? \(\frac{2sinBcosB}{x+1} = \frac{sin B}{x-1} \)? \(2cosB = \frac{x+1}{x-1} \) ----(1)Now, by using the cosine rule,\(cosB = \frac{a^2 + c^2 - b^2}{2ac}\)On substituting the value of the respective variable in the above formula, we get,? \(cosB = \frac{(x+1)^2 + x^2 - (x-1)^2}{2(x+1)x}\)? \(cosB = \frac{(x^2+1 + 2x + x^2 - x^2-1+2x)}{2x^2+2x}\)? \(cosB = \frac{ (4 + x)}{2(x+1)}\) ----(2)On equating (1) and (2), we get,? \(\frac{ (4 + x)}{(x+1)} = \frac{x+1}{x-1}\)? 4x - 4 + x2 - x = x2 + 1 + 2x? x = 5 unitsThe length of the largest side, x + 1 = 5 + 1 = 6 units? 6 units is the required answer.