Question

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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 not a rectangle because it has no right angles
quadrilateral defg is not a rectangle because it has only two right angles
quadrilateral defg is a rectangle because it has four right angles.
quadrilateral defg is not a rectangle because it has only one right angle

Explanation:

Step1: Calculate the slopes of the sides

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

Step2: Check for perpendicularity (product of slopes = - 1 for right - angle)

\(m_{DE}\times m_{EF}=-\frac{4}{3}\times1=-\frac{4}{3}
eq - 1\)
\(m_{EF}\times m_{FG}=1\times(-\frac{3}{4})=-\frac{3}{4}
eq - 1\)
\(m_{FG}\times m_{GD}=-\frac{3}{4}\times1=-\frac{3}{4}
eq - 1\)
\(m_{GD}\times m_{DE}=1\times(-\frac{4}{3})=-\frac{4}{3}
eq - 1\)

Answer:

Quadrilateral \(DEFG\) is not a rectangle because it has no right angles.