The sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one, then the sides of the triangle (in units) are

Step-by-step solution

Let AC = n , AB = n +1, BC = n +2 Largest angle is A and smallest angle is B . A =2 ~B Since A + B + C =180^ 3 ~B + C =180^ aligned & C =180^ -3 ~B & C = (180^ -3 ~B )= 3 ~B aligned By sine rule, aligned & A n+2 = B n = C n+1 & 2 B n+2 = B n = 3 B n+1 & 2 B B n+2 = B n = 3 B-4 ^3 B n+1 & 2 B n+2 = 1 n = 3-4 ^2 B n+1 & B= n+2 2 n , 3-4 ^2 B= n+1 n & 3-4 (1- ^2 B )= n+1 n & -4+4 ( n+2 2 n )^2= n+1 n & -1+ n^2+4 n+4 n^2 = n+1 n & -n^2+n^2+4 n+4=n^2+n & n^2-3 n-4=0 & (n+1)(n-4)=0 & n=-1 or n=4 aligned But n cannot be n