Question

calculator
the coordinates of the vertices of $\triangle def$ are $d(-4, 1)$, $e(3, -1)$, and $f(-1, -4)$
which statement correctly describes whether $\triangle def$ is a right triangle?
$\bigcirc$ $\triangle def$ is a right triangle because $\overline{de}$ is perpendicular to $\overline{ef}$
$\bigcirc$ $\triangle def$ is a right triangle because $\overline{df}$ is perpendicular to $\overline{ef}$
$\bigcirc$ $\triangle def$ is a right triangle because $\overline{de}$ is perpendicular to $\overline{df}$
$\bigcirc$ $\triangle def$ is not a right triangle because no two sides are perpendicular

Explanation:

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

Slope formula: $m=\frac{y_2-y_1}{x_2-x_1}$
$m_{DE}=\frac{-1-1}{3-(-4)}=\frac{-2}{7}$

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

$m_{EF}=\frac{-4-(-1)}{-1-3}=\frac{-3}{-4}=\frac{3}{4}$

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

$m_{DF}=\frac{-4-1}{-1-(-4)}=\frac{-5}{3}$

Step4: Check perpendicularity (product=-1)

$m_{DE} \times m_{EF} = \frac{-2}{7} \times \frac{3}{4} = -\frac{3}{14}
eq -1$
$m_{DF} \times m_{EF} = \frac{-5}{3} \times \frac{3}{4} = -\frac{5}{4}
eq -1$
$m_{DE} \times m_{DF} = \frac{-2}{7} \times \frac{-5}{3} = \frac{10}{21}
eq -1$

Answer:

$\triangle DEF$ is not a right triangle because no two sides are perpendicular.