Question

select the correct answer from each drop - down menu.
the coordinates of a quadrilateral are (2,1), (-1,3), (-5,-3), and (-2,-5).
the quadrilateral is a

because

are parallel and since the product of the slopes of both pairs of segments is - 1,

Explanation:

Step1: Calculate the slopes of the sides

Let the points be \(A(2,1)\), \(B(-1,3)\), \(C(-5,-3)\), \(D(-2,-5)\)
The slope formula is \(m = \frac{y_2 - y_1}{x_2 - x_1}\)
Slope of \(AB\): \(m_{AB}=\frac{3 - 1}{-1 - 2}=\frac{2}{-3}=-\frac{2}{3}\)
Slope of \(BC\): \(m_{BC}=\frac{-3 - 3}{-5+1}=\frac{-6}{-4}=\frac{3}{2}\)
Slope of \(CD\): \(m_{CD}=\frac{-5 + 3}{-2 + 5}=\frac{-2}{3}=-\frac{2}{3}\)
Slope of \(DA\): \(m_{DA}=\frac{1 + 5}{2+2}=\frac{6}{4}=\frac{3}{2}\)

Step2: Analyze the properties

Since \(m_{AB}=m_{CD}=-\frac{2}{3}\) and \(m_{BC}=m_{DA}=\frac{3}{2}\), both pairs of opposite sides are parallel.
Also, \(m_{AB}\times m_{BC}=(-\frac{2}{3})\times(\frac{3}{2})=- 1\), \(m_{CD}\times m_{DA}=(-\frac{2}{3})\times(\frac{3}{2})=-1\)

Answer:

The quadrilateral is a rectangle because both pairs of opposite sides are parallel and since the product of the slopes of both pairs of segments is \(-1\), the adjacent sides are perpendicular