$a)1^2+6^2=c^2$  

$1+36=c^2$  

$c^2=37\mid \sqrt{}$  

$c=\sqrt{37}$  

$6^2+{\left(5+1\right)}^2=d^2$  

$36+36=d^2$  

$d^2=72\mid \sqrt{}$  

$d=\sqrt{72}=\sqrt{36\cdot 2}=6\sqrt{2}$  

 

$b)4^2+6^2=c^2$  

$16+36=c^2$  

$52=c^2\mid \sqrt{}$  

$x=\sqrt{52}=\sqrt{4\cdot 13}=2\sqrt{13}$  

  

$6^2+{\left(5+4\right)}^2=d^2$

$36+81=d^2$  

$117=d^2\mid \sqrt{}$  

$d=\sqrt{117}=\sqrt{9\cdot 13}=3\sqrt{13}$   

 

$c)5^2+6^2=c^2$  

$25+36=c^2$  

$c^2=61\mid \sqrt{}$  

$c=\sqrt{61}$  

 

$6^2+{\left(5+5\right)}^2=d^2$  

$36+{10}^2=d^2$  

$36+100=d^2$  

$d^2=136\mid \sqrt{}$  

$d=\sqrt{136}=\sqrt{4\cdot 34}=2\sqrt{34}$  

 

$d){\left(\sqrt{2}\right)}^2+6^2=c^2$  

$c^2=2+36$  

$c^2=38\mid \sqrt{}$  

$c=\sqrt{38}$  

 

$6^2+{\left(5+\sqrt{2}\right)}^2=d^2$  

$36+5^2+2\cdot 5\cdot \sqrt{2}+{\left(\sqrt{2}\right)}^2=d^2$  

$36+25+10\sqrt{2}+2=d^2$  

$d^2=63+10\sqrt{2}\mid \sqrt{}$  

$d=\sqrt{63+10\sqrt{2}}$