To prove that the vertices form a parallelogram by showing that the opposite sides are parallel, we need to calculate the slopes of the sides NO, OS, SY, and YN.
Let's call the vertices as follows for clarity:
- N = (x1, y1)
- O = (x2, y2)
- S = (x3, y3)
- Y = (x4, y4)
The slope of a line between two points (x1, y1) and (x2, y2) is given by the formula:
\[ \text{slope} = \frac{y2 - y1}{x2 - x1} \]
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Calculate the slope of NO: \[ \text{slope of NO} = \frac{y2 - y1}{x2 - x1} \quad \text{(Insert this value in the first response area)} \]
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Calculate the slope of OS: \[ \text{slope of OS} = \frac{y3 - y2}{x3 - x2} \quad \text{(Insert this value in the second response area)} \]
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Calculate the slope of SY: \[ \text{slope of SY} = \frac{y4 - y3}{x4 - x3} \quad \text{(Insert this value in the third response area)} \]
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Calculate the slope of YN: \[ \text{slope of YN} = \frac{y1 - y4}{x1 - x4} \quad \text{(Insert this value in the fourth response area)} \]
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Final Conclusion: After calculating the slopes, compare them:
- If the slope of NO is equal to the slope of SY, and the slope of OS is equal to the slope of YN, then the opposite sides are parallel.
You can conclude: \[ \text{Is this a parallelogram? Yes.} \quad \text{I know this because the opposite sides of the quadrilateral are parallel.} \]
Make sure to insert the calculated slopes and conclusion into the provided response areas.