Question

a quadrilateral has vertices e(-4, 2), f(4, 7), g(8, 1), and h(0, -4). which statements are true? check all that apply. the slope of eh is - 8/5. the slopes of ef and gh are both 5/8. fg is perpendicular to gh. quadrilateral efgh is a parallelogram because both pairs of opposite sides are parallel. quadrilateral efgh is a rectangle because all angles are right angles.

Explanation:

Step1: Recall slope formula

The slope formula between two points $(x_1,y_1)$ and $(x_2,y_2)$ is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of EH

For points $E(-4,2)$ and $H(0,-4)$, $m_{EH}=\frac{-4 - 2}{0-(-4)}=\frac{-6}{4}=-\frac{3}{2}
eq-\frac{8}{5}$.

Step3: Calculate slope of EF

For points $E(-4,2)$ and $F(4,7)$, $m_{EF}=\frac{7 - 2}{4-(-4)}=\frac{5}{8}$.

Step4: Calculate slope of GH

For points $G(8,1)$ and $H(0,-4)$, $m_{GH}=\frac{-4 - 1}{0 - 8}=\frac{-5}{-8}=\frac{5}{8}$.

Step5: Check perpendicularity

For two lines to be perpendicular, the product of their slopes is - 1. Slope of $FG$ for points $F(4,7)$ and $G(8,1)$ is $m_{FG}=\frac{1 - 7}{8 - 4}=\frac{-6}{4}=-\frac{3}{2}$. Since $m_{FG}\times m_{GH}=-\frac{3}{2}\times\frac{5}{8}
eq - 1$, $FG$ is not perpendicular to $GH$.

Step6: Check parallelogram property

Since $m_{EF}=m_{GH}=\frac{5}{8}$ and we can calculate the slope of $EH$ and $FG$. If $m_{EH}=m_{FG}$, then both pairs of opposite - sides are parallel and the quadrilateral is a parallelogram.

Step7: Check rectangle property

Since $FG$ is not perpendicular to $GH$, not all angles are right - angles, so it is not a rectangle.

Answer:

The slopes of $\overline{EF}$ and $\overline{GH}$ are both $\frac{5}{8}$.
Quadrilateral $EFGH$ is a parallelogram because both pairs of opposite sides are parallel.