Question

Quadrilateral \[ABCD\] has the following vertices: \[A(2,7)\] \[B(8,1)\] \[C(1,-9)\] \[D(-6,-2)\] Is quadrilateral \[ABCD\] a parallelogram, and why? Choose 1 answer: Choose 1 answer: (Choice A) Yes, because \[\overline{AB} \parallel \overline{CD}\], and \[\overline{AB} \perp \overline{AD}\]. A Yes, because \[\overline{AB} \parallel \overline{CD}\], and \[\overline{AB} \perp \overline{AD}\]. (Choice B) Yes, because \[\overline{AB} \parallel \overline{CD}\], and \[\overline{BC} \parallel \overline{AD}\]. B Yes, because \[\overline{AB} \parallel \overline{CD}\], and \[\overline{BC} \parallel \overline{AD}\]. (Choice C) No, because \[\overline{AB}\] is not parallel to \[\overline{CD}\]. C No, because \[\overline{AB}\] is not parallel to \[\overline{CD}\]. (Choice D) No, because \[\overline{BC}\] is not parallel to \[\overline{AD}\]. D No, because \[\overline{BC}\] is not parallel to \[\overline{AD}\].

Answer 1

No, quadrilateral ABCD is not a parallelogram.

To be a parallelogram, a quadrilateral must have two pairs of opposite sides that are parallel. However, in quadrilateral ABCD, only one pair of opposite sides is parallel:

`\rm \overline{AB}\: and \: </p><p> \overline{CD}. \: Sides \: \overline{BC} \: and \: \overline{AD} \: are \: not \: parallel.`

Here is a Python code to calculate the slopes of the sides and check if the opposite sides are parallel:

import matplotlib.pyplot as plt

import numpy as np

# Define the coordinates of the quadrilateral

A = (2, 7)

B = (8, 1)

C = (1, -9)

D = (-6, -2)

# Calculate the slopes of the sides

AB_slope = (B[1] - A[1]) / (B[0] - A[0])

BC_slope = (C[1] - B[1]) / (C[0] - B[0])

CD_slope = (D[1] - C[1]) / (D[0] - C[0])

DA_slope = (A[1] - D[1]) / (A[0] - D[0])

# Check if the opposite sides are parallel

if AB_slope == CD_slope and BC_slope == DA_slope:

print("Yes, quadrilateral ABCD is a parallelogram.")

else:

print("No, quadrilateral ABCD is not a parallelogram.")

Output:

No, quadrilateral ABCD is not a parallelogram.

Therefore, the answer is (C).