Let a, b, c be the lengths of sides of triangle ABC such that a + b 7 = b + c 8 = c + a 9 = k . Then ( A ( ABC ))^2 k ^4 =

Step-by-step solution

aligned & In ABC , & a + b 7 = b + c 8 = c + a 9 k & a + b =7 k ...(i) & b + c =8 k ... (ii) & c + a =9 k ... (iii) & aligned Adding above equations, aligned & 2 a+2 b+2 c=24 k & a+b+c=12 k ... (iv) aligned Solving equations (i), (ii), (iii), (iv) We get, array ll & c =5 k , a =4 k , b =3 k & c ^2= a ^2+ b ^2 array ABC is right angled triangle C =90^ aligned & Area of aligned ABC & = 1 2 ab C & = 1 2 ab 90 & = 1 2 4 k 3 k & =6 k ^2 aligned & Now, [ A ( ABC )]^2 k ^4 = (6 k ^2 )^2 k ^4 = 36 k ^4 k ^4 =36 aligned