Answer:
T= (-4 ,2)
U= (2 ,4)
V= (3 ,1)
W= (-3, -1)
1 . LENGTH OF EACH SIDES
[tex]length \ of \ sides =\sqrt{(x_{2} -x_{1} )^2+(y_{2} -y_{1})^2}[/tex]
[tex]length \ of \ TU = \sqrt{(2--4 )^2+(4-2)^2} = \sqrt{40}[/tex]
[tex]Length \ of \ UV =\sqrt{(3-2 )^2+(1-4)^2} = \sqrt{10}[/tex]
[tex]Length \ of \ VW =\sqrt{(-3-3)^2+(-1-1)^2} = \sqrt{40}[/tex]
[tex]length \ of \ TW =\sqrt{(-3--4 )^2+(-1-2)^2} =\sqrt{10}[/tex]
2. SLOPE OF CONSECUTIVE SIDES
[tex]Slope \ of \ the \ line \ passing \ through \ (x_{1}, y_{1} ) \ and \ (x_{2}, y_{2}) = \frac{y_{2} -y_{1} }{x_{2} -x_{1} }[/tex]
[tex]Slope \ of \ TU = \frac{4-2}{2--4} = \frac{2}{6} =\frac{1}{3}[/tex]
[tex]Slope \ of \ UV = \frac{1-4}{3-2} = -3[/tex]
[tex]Slope \ of \ VW = \frac{-1-1}{-3-3} = \frac{-2}{-6} =\frac{1}{3}[/tex]
[tex]Slope \ of \ TW = \frac{-1-2}{-3--4} = -3[/tex]
[tex]Slope \ of \ lines \ parallel \ to \ each \ other = m_{1}\cdot m_{2} = 1\\Slope\ of \ lines\ perpendicular \ to \ each \ other =m_{1}\cdot m_{2} = -1\\[/tex]
[tex]Slope \ of \ TU * UV = \frac{1}{3}*-3 = -1\\\\Slope \ of \ UV *VW = -3 *\frac{1}{3}= -1\\\\ Slope \ of \ VW * TW = \frac{1}{3}*-3 = -1\\\\Slope \ of \ TW * TU = -3 *\frac{1}{3}= -1\\\\[/tex]
Therefore the consecutive sides are perpendicular to each other.
3. TYPE OF PARALLELOGRAM
The opposite sides are of same length. The consecutive sides are perpendicular to each other. So the parallelogram is a RECTANGLE.
