Question

Hey there Ms or Mr could you please help me out with this assignment? I have to determine if the figure is a parallelogram using the distance formula, slope formula, or midpoint formula. Please help Ms or Mr.Just a heads up this isn't a quiz it's my homework assignment for today is about proving parallelograms & Rectangles on coordinate plane.

Answer 1

Given: The vertices of quadrilateral EFGH, E(-6,2), F(3,8), G(7,2) and H(-2,-4).

The length of a line between two points is given by the equation,

`\begin{gathered} D=\sqrt[]{(x_2-x_1)^2+(y_2-y_1)^2} \\ \text{Here, (x}_{1_{}},y_1)and(x_2,y_2)\text{ are the coordinates of points} \end{gathered}`

Now, the length of side EF of the quadrilateral is,

`\begin{gathered} EF=\sqrt[]{(3-(-6))^2+(8-2)^2} \\ =\sqrt[]{81+36} \\ =\sqrt[]{117} \end{gathered}`

The length of side FG is,

`\begin{gathered} FG=\sqrt[]{(7-3)^2+(2-8)^2} \\ =\sqrt[]{16+36} \\ =\sqrt[]{52} \end{gathered}`

The length of side GH is,

`\begin{gathered} GH=\sqrt[]{(-2-7)^2+(-4-2)^2} \\ =\sqrt[]{117} \end{gathered}`

The length of side HE is,

`\begin{gathered} HE=\sqrt[]{(-2-(-6))^2+(-4-2)^2} \\ =\sqrt[]{52} \end{gathered}`

From above distances, we find that the length of opposite sides EF and GH of quadrilateral EFGH are equal. Similarly, the lengths of another pair of opposite sides FG and HE of quadrilateral EFGH are also equal.

If the pair of opposite sides of a quadrilateral are equal, then the quadrilateral is a rectangle.