Question

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

Explanation:

Step1: Calculate slope of $DE$

The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. For points $D(-4,1)$ and $E(3,-1)$, $m_{DE}=\frac{-1 - 1}{3-(-4)}=\frac{-2}{7}=-\frac{2}{7}$.

Step2: Calculate slope of $EF$

For points $E(3,-1)$ and $F(-1,-4)$, $m_{EF}=\frac{-4 - (-1)}{-1 - 3}=\frac{-3}{-4}=\frac{3}{4}$.

Step3: Calculate slope of $DF$

For points $D(-4,1)$ and $F(-1,-4)$, $m_{DF}=\frac{-4 - 1}{-1-(-4)}=\frac{-5}{3}=-\frac{5}{3}$.

Step4: Check perpendicularity

Two lines with slopes $m_1$ and $m_2$ are perpendicular if $m_1\times m_2=-1$.
$m_{DE}\times m_{EF}=-\frac{2}{7}\times\frac{3}{4}=-\frac{3}{14}
eq - 1$.
$m_{DE}\times m_{DF}=-\frac{2}{7}\times(-\frac{5}{3})=\frac{10}{21}
eq - 1$.
$m_{DF}\times m_{EF}=-\frac{5}{3}\times\frac{3}{4}=-\frac{5}{4}
eq - 1$.

Answer:

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