A triangle with side lengths of a, b and c is a right triangle where a < b < c . Which of the following statements are possible?

Step-by-step solution

(a) a, a+d, a+2 d aligned & a^2+(a+d)^2=(a+2 d)^2 & 2 a^2+d^2+2 a d=a^2+4 d^2+4 a d & a^2-2 a d=3 d^2 & ( a d )^2- 2 a 3 =3 aligned t^2-2 t-3=0 (t-3)(t+1)=0 t=3a=3 d [possible] (b) a=a ; b=a r ; c=a r^2 aligned & c^2=a^2+b^2 & a^2 r^4=a^2 (1+r^2 ) aligned r^2 t^2-t-1=0 array ll r^2= 1+ 5 2 >1 [possible] array aligned & a, b, c A.P. & b= 2 a c a+c & c^2=a^2+ 4 a^2 c^2 (a+c)^2 & a^4+2 a^3 c+4 a^2 c^2-2 a c^3-c^4=0 aligned Divide by a^2 c^2 aligned & ( a^2 c^2 - c^2 a^2 )+2 ( a c - c a )+4=0 & a c - c a =t ; t 0 array