To determine if the quadrilateral RICO is a parallelogram, you can either check if the opposite sides are parallel (by comparing slopes), or if the opposite sides are of equal length (by calculating distances).
Let's analyze both methods:
1. Checking Slopes
The coordinates of the points are:
- R (0, 1)
- I (2, -2)
- C (3, 1)
- O (1, 4)
Calculating the Slopes:
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Slope of line RI: \[ \text{slope of RI} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - 1}{2 - 0} = \frac{-3}{2} \]
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Slope of line CO: \[ \text{slope of CO} = \frac{1-4}{3-1} = \frac{-3}{2} \]
So, lines RI and CO are parallel.
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Slope of line IC: \[ \text{slope of IC} = \frac{1 - (-2)}{3 - 2} = \frac{3}{1} = 3 \]
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Slope of line OR: \[ \text{slope of OR} = \frac{4 - 1}{1 - 0} = \frac{3}{1} = 3 \]
So, lines IC and OR are also parallel.
Since both pairs of opposite sides (RI and CO, IC and OR) are parallel, RICO is a parallelogram.
2. Checking Distances
Calculating the Distances:
-
Distance of RI: \[ d_{RI} = \sqrt{(2 - 0)^2 + (-2 - 1)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
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Distance of CO: \[ d_{CO} = \sqrt{(3 - 1)^2 + (1 - 4)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]
Both RI and CO have the same distance.
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Distance of IC: \[ d_{IC} = \sqrt{(3 - 2)^2 + (1 - (-2))^2} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]
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Distance of OR: \[ d_{OR} = \sqrt{(1 - 0)^2 + (4 - 1)^2} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]
Both IC and OR also have the same distance.
Conclusion
Both properties (parallel sides and congruent sides) indicate that RICO is a parallelogram.
The following responses are correct:
- Yes. I found the slopes of RI and CO and found that they had the same slope. I then found the slopes of IC and OR and they also had the same slope. Since they have the same slopes, it means that the opposite sides are parallel to each other. This proves that RICO is a parallelogram.
- Yes. I found the distance of RI and CO and found that they are the same distance. I then found the distance of IC and OR and they are also the same distance. Since they are the same distance, it means that the opposite sides are congruent to each other. This proves that RICO is a parallelogram.
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