Question

lydia graped $delta def$ at the coordinates $d (-2, -1)$, $e (-2, 2)$, and $f (0, 0)$. she thinks $delta def$ is a right triangle. is lydias assertion correct?
yes, the slopes of $overline{ef}$ and $overline{df}$ are opposite reciprocals.
yes, the slopes of $overline{ef}$ and $overline{de}$ are the same.
no, the slopes of $overline{ef}$ and $overline{df}$ are not opposite reciprocals.
no, the slopes of $overline{ef}$ and $overline{de}$ are not the same.

Explanation:

Step1: Calculate slope of $\overline{DE}$

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_{DE}=\frac{2-(-1)}{-2-(-2)}=\frac{3}{0}$ (undefined, vertical line)

Step2: Calculate slope of $\overline{EF}$

$m_{EF}=\frac{0-2}{0-(-2)}=\frac{-2}{2}=-1$

Step3: Calculate slope of $\overline{DF}$

$m_{DF}=\frac{0-(-1)}{0-(-2)}=\frac{1}{2}$

Step4: Check perpendicular slopes

Perpendicular slopes are opposite reciprocals. $\overline{DE}$ is vertical, so a horizontal line would be perpendicular, but neither $\overline{EF}$ nor $\overline{DF}$ is horizontal. Also, $-1 \times \frac{1}{2}
eq -1$. No two sides have slopes that are opposite reciprocals.

Answer:

No, the slopes of $\overline{EF}$ and $\overline{DF}$ are not opposite reciprocals