Question

the coordinates of the vertices of quadrilateral defg are d(-3, -1), e(-6, 3), f(3, 12), and g(7, 9). which statement correctly describes whether quadrilateral defg is a rectangle? quadrilateral defg is a rectangle because it has four right angles. quadrilateral defg is not a rectangle because it has only one right angle. quadrilateral defg is not a rectangle because it has only two right angles. quadrilateral defg is not a rectangle because it has no right angles.

Explanation:

Step1: Calculate slope of sides

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
Slope of $DE$: $m_{DE}=\frac{3-(-1)}{-6 - (-3)}=\frac{4}{-3}=-\frac{4}{3}$.
Slope of $EF$: $m_{EF}=\frac{12 - 3}{3-(-6)}=\frac{9}{9} = 1$.
Slope of $FG$: $m_{FG}=\frac{9 - 12}{7 - 3}=\frac{-3}{4}=-\frac{3}{4}$.
Slope of $GD$: $m_{GD}=\frac{-1 - 9}{-3 - 7}=\frac{-10}{-10}=1$.

Step2: Check perpendicularity

Two lines are perpendicular if the product of their slopes is - 1.
$m_{DE}\times m_{EF}=-\frac{4}{3}\times1=-\frac{4}{3}
eq - 1$.
$m_{EF}\times m_{FG}=1\times(-\frac{3}{4})=-\frac{3}{4}
eq - 1$.
$m_{FG}\times m_{GD}=-\frac{3}{4}\times1=-\frac{3}{4}
eq - 1$.
$m_{GD}\times m_{DE}=1\times(-\frac{4}{3})=-\frac{4}{3}
eq - 1$.
Since no two - adjacent sides are perpendicular, there are no right - angles.

Answer:

Quadrilateral $DEFG$ is not a rectangle because it has no right angles.