Question

1. the vertices of a quadrilateral are (-3, 3), (4, 5), (6, -1), and (-1, -3). determine if the quadrilateral is a parallelogram.

Explanation:

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<reasoning>

Calculate slopes of opposite sides

\[

$$\begin{aligned} &A(-3, 3), \quad B(4, 5), \quad C(6, -1), \quad D(-1, -3)\\ &m_{AB} = \frac{5 - 3}{4 - (-3)} = \frac{2}{7}\\ &m_{CD} = \frac{-3 - (-1)}{-1 - 6} = \frac{-2}{-7} = \frac{2}{7}\\ &m_{BC} = \frac{-1 - 5}{6 - 4} = \frac{-6}{2} = -3\\ &m_{DA} = \frac{3 - (-3)}{-3 - (-1)} = \frac{6}{-2} = -3 \end{aligned}$$

\]

Compare slopes of opposite sides

\[

$$\begin{aligned} &m_{AB} = m_{CD} = \frac{2}{7} \implies AB \parallel CD\\ &m_{BC} = m_{DA} = -3 \implies BC \parallel DA \end{aligned}$$

\]

Determine quadrilateral type

\[
\text{Both pairs of opposite sides are parallel } \implies ABCD \text{ is a parallelogram.}
\]
</reasoning>

<answer>
Yes, the quadrilateral is a parallelogram because both pairs of opposite sides are parallel (the slopes of opposite sides are equal: \(m_{AB} = m_{CD} = \frac{2}{7}\) and \(m_{BC} = m_{DA} = -3\)).
</answer>

<plot>
{
"elements": [
{
"type": "point",
"params": [[-3, 3]],
"properties": {
"name": "A(-3,3)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[4, 5]],
"properties": {
"name": "B(4,5)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[6, -1]],
"properties": {
"name": "C(6,-1)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[-1, -3]],
"properties": {
"name": "D(-1,-3)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "polygon",
"params": [[-3, 3], [4, 5], [6, -1], [-1, -3]],
"properties": {
"fillColor": "rgba(140, 85, 242, 0.2)",
"strokeColor": "#8C55F2",
"strokeWidth": 2
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Answer:

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<reasoning>

Calculate slopes of opposite sides

\[

$$\begin{aligned} &A(-3, 3), \quad B(4, 5), \quad C(6, -1), \quad D(-1, -3)\\ &m_{AB} = \frac{5 - 3}{4 - (-3)} = \frac{2}{7}\\ &m_{CD} = \frac{-3 - (-1)}{-1 - 6} = \frac{-2}{-7} = \frac{2}{7}\\ &m_{BC} = \frac{-1 - 5}{6 - 4} = \frac{-6}{2} = -3\\ &m_{DA} = \frac{3 - (-3)}{-3 - (-1)} = \frac{6}{-2} = -3 \end{aligned}$$

\]

Compare slopes of opposite sides

\[

$$\begin{aligned} &m_{AB} = m_{CD} = \frac{2}{7} \implies AB \parallel CD\\ &m_{BC} = m_{DA} = -3 \implies BC \parallel DA \end{aligned}$$

\]

Determine quadrilateral type

\[
\text{Both pairs of opposite sides are parallel } \implies ABCD \text{ is a parallelogram.}
\]
</reasoning>

<answer>
Yes, the quadrilateral is a parallelogram because both pairs of opposite sides are parallel (the slopes of opposite sides are equal: \(m_{AB} = m_{CD} = \frac{2}{7}\) and \(m_{BC} = m_{DA} = -3\)).
</answer>

<plot>
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"elements": [
{
"type": "point",
"params": [[-3, 3]],
"properties": {
"name": "A(-3,3)",
"size": 4,
"color": "#8C55F2",
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},
{
"type": "point",
"params": [[4, 5]],
"properties": {
"name": "B(4,5)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
}
},
{
"type": "point",
"params": [[6, -1]],
"properties": {
"name": "C(6,-1)",
"size": 4,
"color": "#8C55F2",
"withLabel": true
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{
"type": "point",
"params": [[-1, -3]],
"properties": {
"name": "D(-1,-3)",
"size": 4,
"color": "#8C55F2",
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},
{
"type": "polygon",
"params": [[-3, 3], [4, 5], [6, -1], [-1, -3]],
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