57.
a)
[tex]|AB|:y=ax+b\\\begin{cases}3=2a+b\\1=5a+b\end{cases}\\\begin{cases}3=2a+b\\-1=-5a-b\end{cases}\\\\2=-3a\\a=\cfrac{2}{-3}=-\cfrac{2}{3}[/tex]
[tex]|AC|:y=ax+b\\\begin{cases}3=2a+b\\4=5a+b\end{cases}\\\begin{cases}3=2a+b\\-4=-5a-b\end{cases}\\\\-1=-3a\\a=\cfrac{-1}{-3}=\cfrac{1}{3}[/tex]
[tex]|AD|:y=ax+b\\\begin{cases}3=2a+b\\5=5a+b\end{cases}\\\begin{cases}3=2a+b\\-5=-5a-b\end{cases}\\\\-2=-3a\\a=\cfrac{-2}{-3}=\cfrac{2}{3}[/tex]
b)
[tex]a=-\cfrac{2}{3}\\3=2a+b\\b=-2a+3=1\cfrac{1}{3}+3=4\cfrac{1}{3}\\y=-\cfrac{2}{3}x+4\cfrac{1}{3}[/tex]
[tex]P_{OX}:\begin{array}{c}y=0\\0=-\cfrac{2}{3}x+4\cfrac{1}{3}\\\cfrac{2}{3}x=4\cfrac{1}{3}\\x=6\cfrac{1}{2}\end{array}\\a_\Delta=6\cfrac{1}{2}[/tex]
[tex]P_{OY}:\begin{array}{c}x=0\\y=-\cfrac{2}{3}*0+4\cfrac{1}{3}\\y=4\cfrac{1}{3}\end{array}\\h_\Delta=4\cfrac{1}{3}[/tex]
[tex]S_\Delta=\cfrac{ah}{2}=\cfrac{6\frac{1}{2}*4\frac{1}{3}}{2}=\cfrac{\frac{169}{6}}{2}=\cfrac{169}{12}=14\cfrac{1}{12}[/tex]
c)
[tex]a=\cfrac{1}{3}\\3=2a+b\\b=-2a+3=-\cfrac{2}{3}+3=2\cfrac{1}{3}\\y=\cfrac{1}{3}x+2\cfrac{1}{3}\\\cfrac{1}{3}x+2\cfrac{1}{3}<0\\\\x+7<0\\x<-7\\x\in(-\infty;-7)[/tex]
58.
a)
[tex]|AB|:y=ax+b\\\begin{cases}2=-3a+b\\-6=a+b\end{cases}\\\begin{cases}2=-3a+b\\6=-a-b\end{cases}\\\\8=-4a\\a=\cfrac{8}{-4}=-2\\2=-3a+b\\b=3a+2=-6+2=-4\\y=-2x-4[/tex]
[tex]|AC|:y=ax+b\\\begin{cases}2=-3a+b\\6=9a+b\end{cases}\\\begin{cases}2=-3a+b\\-6=-9a-b\end{cases}\\\\-4=-12a\\a=\cfrac{-4}{-12}=\cfrac{1}{3}\\2=-3a+b\\b=3a+2=1+2=3\\y=\cfrac{1}{3}x+3[/tex]
[tex]|BC|:y=ax+b\\\begin{cases}-6=a+b\\6=9a+b\end{cases}\\\begin{cases}-6=a+b\\-6=-9a-b\end{cases}\\\\-12=-8a\\a=\cfrac{-12}{-8}=1.5\\-6=a+b\\b=-a-6=-1.5-6=-7.5\\y=1.5x-7.5[/tex]
b)
[tex]|AB|:y=ax+b\\\begin{cases}-3=-2a+b\\3=6a+b\end{cases}\\\begin{cases}-3=-2a+b\\-3=-6a-b\end{cases}\\\\-6=-8a\\a=\cfrac{-6}{-8}=\cfrac{3}{4}\\-3=-2a+b\\b=2a-3=1.5-3=-1.5\\y=\cfrac{3}{4}x-1.5[/tex]
[tex]|BC|:y=ax+b\\\begin{cases}3=6a+b\\4=-a+b\end{cases}\\\begin{cases}3=6a+b\\-4=a-b\end{cases}\\\\-1=7a\\a=\cfrac{-1}{7}=-\cfrac{1}{7}\\3=6a+b\\b=-6a+3=\cfrac{6}{7}+3=3\cfrac{6}{7}\\y=-\cfrac{1}{7}x+3\cfrac{6}{7}[/tex]
[tex]|CD|:y=ax+b\\\begin{cases}4=-a+b\\1=-5a+b\end{cases}\\\begin{cases}4=-a+b\\-1=5a-b\end{cases}\\\\3=4a\\a=\cfrac{3}{4}\\4=-a+b\\b=a+4=\cfrac{3}{4}+4=4\cfrac{3}{4}\\y=\cfrac{3}{4}x+4\cfrac{3}{4}[/tex]
[tex]|DA|:y=ax+b\\\begin{cases}1=-5a+b\\-3=-2a+b\end{cases}\\\begin{cases}1=-5a+b\\3=2a-b\end{cases}\\\\4=-3a\\a=\cfrac{4}{-3}=-\cfrac{4}{3}\\1=-5a+b\\b=5a+1=-\cfrac{20}{3}+1=-\cfrac{17}{3}\\y=-\cfrac{4}{3}x-\cfrac{17}{3}[/tex]
Jest to trapez prostokątny ponieważ prosta leżąca na odcinku [tex]|DA|[/tex] ma współczynnik kierunkowy przeciwny i odwrotny do prostych leżących na odcinkach [tex]|AB|[/tex] i [tex]|CD|[/tex], co jest warunkiem prostych prostopadłych.
