Question

When the coordinates (1, 1), (4, 4), (7, 1), and (4, −2) are joined, which shape is formed?

Answer 1

A square is made by those coordinates

Answer 2

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.

    $AB=\sqrt{(4-1)^2+(4-1)^2}\\\\AB=\sqrt{9+9}\\\\AB=\sqrt{18}\\\\AB=3\sqrt{2}\\\\BC=\sqrt{(7-4)^2+(1-4)^2}\\\\BC=\sqrt{9+9}\\\\BC=\sqrt{18}\\\\BC=3\sqrt{2}\\\\CD=\sqrt{(7-4)^2+(1+2)^2}\\\\CD=\sqrt{9+9}\\\\CD=3\sqrt{2}\\\\AD=\sqrt{(4-1)^2+(-2-1)^2}\\\\AD=\sqrt{9+9}\\\\AD=3\sqrt{2}$

As all sides are equal, so this can be Square or Rhombus.

   Slope between two points (1, 1), and (4, 4) is

        $m_{1}=\frac{4-1}{4-1}\\\\=1\\\\\text{Between }(4,4) \text{and} (7,1)\\\\m_{2}=\frac{4-1}{4-7}\\\\m_{2}=\frac{-1}{3}\\\\m_{1}*m_{2}=3*\frac{-1}{3}\\\\m_{1}*m_{2}=-1$

⇒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