Question

is fghi a trapezoid? justify your answer. yes, $overline{fi}$ is congruent to $overline{gh}$. yes, $overline{fg}$ is parallel to... no, none of the sides are congruent. no, none of the sides are parallel.

Explanation:

A trapezoid is defined as a quadrilateral with at least one pair of parallel sides. To confirm parallelism, calculate the slope of each side using the formula $m=\frac{y_2-y_1}{x_2-x_1}$:

1. Slope of $\overline{FG}$: Points $F(5,-1)$ and $G(3,4)$: $m_{FG}=\frac{4-(-1)}{3-5}=\frac{5}{-2}=-\frac{5}{2}$
2. Slope of $\overline{GH}$: Points $G(3,4)$ and $H(-3,2)$: $m_{GH}=\frac{2-4}{-3-3}=\frac{-2}{-6}=\frac{1}{3}$
3. Slope of $\overline{HI}$: Points $H(-3,2)$ and $I(-2,-5)$: $m_{HI}=\frac{-5-2}{-2-(-3)}=\frac{-7}{1}=-7$
4. Slope of $\overline{FI}$: Points $F(5,-1)$ and $I(-2,-5)$: $m_{FI}=\frac{-5-(-1)}{-2-5}=\frac{-4}{-7}=\frac{4}{7}$

None of the slopes are equal, meaning no sides are parallel. A quadrilateral with no parallel sides cannot be a trapezoid.

Answer:

No, none of the sides are parallel.