Answer with explanation:
The coordinates of four points are A(1, 1), B(4, 4), C(7, 1), and D(4, −2).
Plotting the four points in the two dimensional Coordinate plane
When you will plot these points you will get the shape of a Quadrilateral.
[tex]AB=\sqrt{(4-1)^2+(4-1)^2}\\\\AB=\sqrt{9+9}\\\\AB=\sqrt{18}\\\\AB=3\sqrt{2}\\\\BC=\sqrt{(7-4)^2+(1-4)^2}\\\\BC=\sqrt{9+9}\\\\BC=\sqrt{18}\\\\BC=3\sqrt{2}\\\\CD=\sqrt{(7-4)^2+(1+2)^2}\\\\CD=\sqrt{9+9}\\\\CD=3\sqrt{2}\\\\AD=\sqrt{(4-1)^2+(-2-1)^2}\\\\AD=\sqrt{9+9}\\\\AD=3\sqrt{2}[/tex]
As all sides are equal, so this can be Square or Rhombus.
Slope between two points (1, 1), and (4, 4) is
[tex]m_{1}=\frac{4-1}{4-1}\\\\=1\\\\\text{Between }(4,4) \text{and} (7,1)\\\\m_{2}=\frac{4-1}{4-7}\\\\m_{2}=\frac{-1}{3}\\\\m_{1}*m_{2}=3*\frac{-1}{3}\\\\m_{1}*m_{2}=-1[/tex]
⇒Prodct of slopes is -1, showing that lines which are sides of Quadrilateral are Perpendicular.
⇒As angle between sides measures 90°.So, the Quadrilateral is Square.
→Square
