Question

consider quadrilateral efgh. the slope of ef is 2/3. which statements are true? check all that apply. eh is parallel to fg. ef is perpendicular to eh. hg is neither parallel nor perpendicular to fg. quadrilateral efgh is a rectangle because it is a parallelogram with four right angles. quadrilateral efgh is a trapezoid because it has exactly one pair of parallel opposite sides.

Explanation:

Step1: Recall slope - parallel and perpendicular rules

Parallel lines have equal slopes. Perpendicular lines have slopes whose product is - 1. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$.

Step2: Calculate slope of EH

$E(-8,5)$ and $H(0, - 7)$. $m_{EH}=\frac{-7 - 5}{0+8}=\frac{-12}{8}=-\frac{3}{2}$. Since $m_{EF}=\frac{2}{3}$ and $m_{EF}\times m_{EH}=\frac{2}{3}\times(-\frac{3}{2})=-1$, EF is perpendicular to EH.

Step3: Calculate slope of FG

$F(-2,9)$ and $G(5,-3)$. $m_{FG}=\frac{-3 - 9}{5 + 2}=\frac{-12}{7}$.

Step4: Calculate slope of HG

$H(0,-7)$ and $G(5,-3)$. $m_{HG}=\frac{-3+7}{5 - 0}=\frac{4}{5}$. Since $m_{HG}
eq m_{FG}$ and $m_{HG}\times m_{FG}
eq - 1$, HG is neither parallel nor perpendicular to FG.

Step5: Check for parallel sides

$m_{EH}=-\frac{3}{2}$, $m_{FG}=-\frac{12}{7}$, so EH is not parallel to FG.

Step6: Check for rectangle or trapezoid

Since there is no pair of parallel sides, it is not a rectangle (which is a parallelogram with four right - angles) or a trapezoid (which has exactly one pair of parallel opposite sides).

Answer:

EF is perpendicular to EH.
HG is neither parallel nor perpendicular to FG.