Question

given quadrilateral ABCD with vertices at A(-2,3), B(1,6), C(5,2), and D(2,-1); determine what type of quadrilateral ABCD is.

Answer 1

Determine each point in cartesian plane. Then connect the points. It will show you a rectangle.

Answer: rectangle

Answer 2

Answer: The quadrilateral ABCD is a RECTANGLE.

Step-by-step explanation:  Given that the co-ordinates of the vertices of quadrilateral ABCD are A(-2,3), B(1,6), C(5,2), and D(2,-1).

We are to find the type of quadrilateral ABCD.

The lengths of the sides of quadrilateral ABCD are calculated using distance formula, as follows:

$AB=\sqrt{(1+2)^2+(6-3)^2}=\sqrt{9+9}=3\sqrt2,\\\\BC=\sqrt{(5-1)^2+(2-6)^2}=\sqrt{16+16}=4\sqrt2,\\\\CD=\sqrt{(2-5)^2+(-1-2)^2}=\sqrt{9+9}=3\sqrt2,\\\\DA=\sqrt{(-2-2)^2+(3+1)^2}=\sqrt{16+16}=4\sqrt2.$

So, AB = CD  and  BC = DA.

Now, the slopes of the sides AB, BC, CD and DA are calculated as follows:

$\textup{slope of AB, m}=\dfrac{6-3}{1+2}=\dfrac{3}{3}=1,\\\\\\\textup{slope of BC, n}=\dfrac{2-6}{5-1}=\dfrac{-4}{4}=-1,\\\\\\\textup{slope of CD, o}=\dfrac{-1-2}{2-5}=\dfrac{-3}{-3}=1,\\\\\\\textup{slope of DA, p}=\dfrac{3+1}{-2-2}=\dfrac{4}{-4}=-1.$

This implies that

m = o, n = p and m × n = n × o = o × p = p × m = -1.

So, the opposite sides are parallel and adjacent sides are perpendicular.

Therefore, the opposite sides of the quadrilateral ABCD are parallel and congruent. Also, the adjacent sides are perpendicular.

Thus, quadrilateral ABCD is a RECTANGLE.