Questions: Choose the best selection for the quadrilateral with vertices at the following points: (0,0),(-2,3),(7,0),(5,3) Hint: Start by graphing the points. Distance Formula: d=sqrt((x2-x1)^2+(y2-y1)^2) A. Trapezoid B. Rectangle C. Parallelogram D. Rhombus

Choose the best selection for the quadrilateral with vertices at the following points:
(0,0),(-2,3),(7,0),(5,3)

Hint: Start by graphing the points.
Distance Formula: d=sqrt((x2-x1)^2+(y2-y1)^2)
A. Trapezoid
B. Rectangle
C. Parallelogram
D. Rhombus

Transcript text: Choose the best selection for the quadrilateral with vertices at the following points: \[ (0,0),(-2,3),(7,0),(5,3) \] Hint: Start by graphing the points. Distance Formula: $d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$ A. Trapezoid B. Rectangle C. Parallelogram D. Rhombus

Solution

Solution Steps

To determine the type of quadrilateral, we need to calculate the distances between each pair of vertices to find the lengths of the sides and diagonals. This will help us identify the properties of the quadrilateral.

Calculate the distances between each pair of vertices using the distance formula.
Compare the lengths of the sides and diagonals to determine if the quadrilateral is a trapezoid, rectangle, parallelogram, or rhombus.

Step 1: Calculate Distances

We have the vertices of the quadrilateral at the points $ (0, 0) $, $ (-2, 3) $, $ (7, 0) $, and $ (5, 3) $. Using the distance formula $ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $, we calculate the following distances:

$ d_1 = \sqrt{((-2) - 0)^2 + (3 - 0)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6056 $
$ d_2 = \sqrt{(7 - (-2))^2 + (0 - 3)^2} = \sqrt{81 + 9} = \sqrt{90} \approx 9.4868 $
$ d_3 = \sqrt{(5 - (-2))^2 + (3 - 3)^2} = \sqrt{49} = 7 $
$ d_4 = \sqrt{(7 - 5)^2 + (0 - 3)^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6056 $
$ d_5 = \sqrt{(7 - 0)^2 + (0 - 0)^2} = \sqrt{49} = 7 $
$ d_6 = \sqrt{(5 - (-2))^2 + (3 - 3)^2} = \sqrt{49} = 7 $

Step 2: Analyze Properties

From the calculated distances:

$ d_1 \approx 3.6056 $
$ d_2 \approx 9.4868 $
$ d_3 \approx 3.6056 $
$ d_4 \approx 5.8309 $
$ d_5 = 7 $
$ d_6 = 7 $

We observe that:

$ d_1 \neq d_2 $
$ d_1 = d_3 $
$ d_5 = d_6 $

Since $ d_1 $ and $ d_3 $ are equal, and $ d_5 $ and $ d_6 $ are equal, but $ d_1 $ and $ d_2 $ are not equal, the quadrilateral does not have all sides equal (not a rhombus) and does not have opposite sides equal (not a parallelogram). However, it has one pair of opposite sides that are equal, indicating that it is a trapezoid.