To answer this question, we will use the following formulas for the slope and the distance with two given points (x₁,y₁) and (x₂,y₂):
[tex]\begin{gathered} \text{slope}=\frac{y_2-y_1}{x_2-x_1}, \\ \text{distance}=\sqrt[]{(y_2-y_1)^2+(x_2-x_1)^2}. \end{gathered}[/tex]
Applying the above formula for the slope we get:
[tex]\begin{gathered} \text{slopeGH}=\frac{6-4}{-7-(-1)}=-\frac{1}{3}, \\ \text{slopeHI}=\frac{4-7}{-1-0}=3, \\ \text{slopeIJ}=\frac{7-9}{0-(-6)}=-\frac{1}{3}, \\ \text{slopeJG}=\frac{9-6}{-6-(-7)}=3\text{.} \end{gathered}[/tex]
Therefore, GH and IJ are parallel and both are perpendicular to HI and JG. Also, HI and JG are parallel.
Using the formula for the distance we get:
[tex]\begin{gathered} \text{lengthGH}=\sqrt[]{(6-4)^2+(-7-(-1))^2}=\sqrt[]{4+36}=\sqrt[]{40}, \\ \text{lengthHI}=\sqrt[]{(4-7)^2+(-1-0)^2}=\sqrt[]{9+1}=\sqrt[]{10}, \\ \text{lengthIJ}=\sqrt[]{(7-9)^2+(0-(-6))^2}=\sqrt[]{4+36}=\sqrt[]{40}, \\ \text{lengthJG}=\sqrt[]{(9-6)^2+(-6-\mleft(-7\mright))^2}=\sqrt[]{9+1}=\sqrt[]{10}\text{.} \end{gathered}[/tex]
Therefore, the parallel sides have the same length.
Answer:
[tex]\begin{gathered} \text{slopeGH}=-\frac{1}{3}, \\ \text{slopeHI}=3, \\ \text{slopeIJ}=-\frac{1}{3}, \\ \text{slopeJG}=3\text{.} \end{gathered}[/tex]
[tex]\begin{gathered} \text{lengthGH}=\sqrt[]{40}, \\ \text{lengthHI}=\sqrt[]{10}, \\ \text{lengthIJ}=\sqrt[]{40}, \\ \text{lengthJG}=\sqrt[]{10}\text{.} \end{gathered}[/tex]
The quadrilateral is a rectangle.
