[tex]|AB|=\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2}\\\\a)\\\\ A=(-5:-1)\ \ \ B=( 5;-1)\\\\|AB|=\sqrt{(-5+3)^2+(-1+1)^2}=\sqrt{(-2)^2+(0)^2} =\sqrt{ 4}=2\\\\b)\\\\ A=(5;-6\frac{1}{2})\ \ \ B=(-7;-1\frac{1}{2})\\\\|AB|=\sqrt{(-7-5)^2+(-1\frac{1}{2}+6\frac{1}{2} )^2}=\sqrt{(-12)^2+5^2} =\\\\=\sqrt{ 144+25}=\sqrt{169}=13[/tex]
[tex]c)\\\\ A=( \frac{1}{3}:- \frac{2}{3})\ \ \ B=( \frac{1}{2};-1)\\\\|AB|=\sqrt{( \frac{1}{2}-\frac{1}{3})^2+(-1+ \frac{2}{3})^2}=\sqrt{( \frac{3}{6}-\frac{2}{6})^2+( -\frac{1}{3})^2} =\sqrt{ (\frac{1}{6})^2+\frac{1}{9} }=\\\\=\sqrt{\frac{5}{36}=} \sqrt{\frac{1}{36}+\frac{4}{36}}=\sqrt{\frac{5}{36}}=\frac{\sqrt{5}}{\sqrt{36}}=\frac{\sqrt{5}}{6}[/tex]
[tex]D)\\\\A=(3+ \sqrt{3} ; \sqrt{7})\ \ \ B=( \sqrt{3};- 4+\sqrt{7})\\\\|AB|=\sqrt{( \sqrt{3}-3-\sqrt{3})^2+(-4+\sqrt{7}-\sqrt{7} )^2}= \sqrt{(-3)^2+(-4)^2}=\sqrt{9+16}=\sqrt{25}=5[/tex]
