Let (x+y):(y+z):(z+x)=3:5:6 Set x+y=3k,y+z=5k,z+x=6k. Add all three: (x+y)+(y+z)+(z+x)=2(x+y+z) Given x+y+z=14, so: 2(14)=28=(3+5+6)k=14k. Thus, k=2. Now compute each pair: x+y=3k=6 y+z=5k=10 z+x=6k=12 Using x+y+z=14 : z=14−(x+y)=14−6=8 x=14−(y+z)=14−10=4 y=14−(z+x)=14−12=2 Now compute: x2+y2+z2=42+22+82=16+4+64=84
