To classify the shape formed by the vertices U(0, 2), N(4, 7), I(6, 3), and T(3, -2), we will follow these steps:
- Calculate the lengths of each side.
- Calculate the slopes of each side to check for parallelism.
- Check for right angles.
- Check for properties of rhombuses and squares.
Step 1: Calculate Lengths of Each Side
Let’s calculate the lengths of the segments UN, NI, IT, and TU using the distance formula, which is given by: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
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Length of UN: \[ UN = \sqrt{(4 - 0)^2 + (7 - 2)^2} = \sqrt{16 + 25} = \sqrt{41} \]
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Length of NI: \[ NI = \sqrt{(6 - 4)^2 + (3 - 7)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]
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Length of IT: \[ IT = \sqrt{(3 - 6)^2 + (-2 - 3)^2} = \sqrt{9 + 25} = \sqrt{34} \]
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Length of TU: \[ TU = \sqrt{(0 - 3)^2 + (2 + 2)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
Step 2: Calculate Slopes of Each Side
Next, we calculate the slopes to find out whether opposite sides are parallel.
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Slope of UN: \[ \text{slope}_{UN} = \frac{7 - 2}{4 - 0} = \frac{5}{4} \]
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Slope of NI: \[ \text{slope}_{NI} = \frac{3 - 7}{6 - 4} = \frac{-4}{2} = -2 \]
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Slope of IT: \[ \text{slope}_{IT} = \frac{-2 - 3}{3 - 6} = \frac{-5}{-3} = \frac{5}{3} \]
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Slope of TU: \[ \text{slope}_{TU} = \frac{2 - (-2)}{0 - 3} = \frac{4}{-3} = -\frac{4}{3} \]
Step 3: Check for Parallelism
We check if opposite sides are parallel by comparing their slopes:
- UN and IT: Slopes \( \frac{5}{4} \) and \( \frac{5}{3} \) are not equal.
- NI and TU: Slopes \( -2 \) and \( -\frac{4}{3} \) are not equal.
Since no pairs of opposite sides are parallel, it is not a parallelogram, rectangle, or rhombus.
Step 4: Check for Other Properties
Since it is not a parallelogram, we can check if it is a quadrilateral:
- All four sides have different lengths ( \( \sqrt{41}, 2\sqrt{5}, \sqrt{34}, 5 \)), so it can't be a rhombus or square.
- The angles are not right angles as no perpendicular slopes (opposite products yielding \(-1\)) are found.
Conclusion
The shape defined by U, N, I, and T is a quadrilateral because it has four sides, but it does not meet the criteria to be classified further as a parallelogram, rectangle, rhombus, or square.