The lengths of sides of a triangle are three consecutive natural numbers and its largest angle is twice the smallest one. Then the length of the sides of the triangle (in units) are

Step-by-step solution

Let a, a+1, a+2 be the sides of the triangle and A , B , C be the angles opposite to them respectively. According to the given condition, C =2 ~A aligned C & = 2 A C & =2 A A aligned Note that A a = C a+2 =k aligned & array l A=k a and C=k(a+2) Also, A= (a+1)^2+(a+2)^2-a^2 2(a+1)(a+2) = a^2+2 a+1+a^2+4 a+4-a^2 2 (a^2+3 a+2 ) = a^2+6 a+5 2 (a^2+3 a+2 ) (i) k(a+2)=2 k a a^2+6 a+5 2 (a^2+3 a+2 ) a+2= a (a^2+6 a+5 ) (a^2+3 a+2 ) (a+2) (a^2+3 a+2 )=a^3+6 a^2+5 a array aligned array ll & a^3+5 a^2+8 a+4=a^3+6 a^2+5 a & a