Given the vertices, determine the most specific classification: Quadrilateral, parallelogram, rectangle, rhombus, or square. U(0,2), N(4,7), I(6,3), T(3,-2)

1 answer

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:

  1. Calculate the lengths of each side.
  2. Calculate the slopes of each side to check for parallelism.
  3. Check for right angles.
  4. 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: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

  • Length of UN: \[ UN = \sqrt{(4 - 0)^2 + (7 - 2)^2} = \sqrt{16 + 25} = \sqrt{41} \]

  • Length of NI: \[ NI = \sqrt{(6 - 4)^2 + (3 - 7)^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5} \]

  • Length of IT: \[ IT = \sqrt{(3 - 6)^2 + (-2 - 3)^2} = \sqrt{9 + 25} = \sqrt{34} \]

  • 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.

  • Slope of UN: \[ \text{slope}_{UN} = \frac{7 - 2}{4 - 0} = \frac{5}{4} \]

  • Slope of NI: \[ \text{slope}_{NI} = \frac{3 - 7}{6 - 4} = \frac{-4}{2} = -2 \]

  • Slope of IT: \[ \text{slope}_{IT} = \frac{-2 - 3}{3 - 6} = \frac{-5}{-3} = \frac{5}{3} \]

  • 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:

  1. All four sides have different lengths ( \( \sqrt{41}, 2\sqrt{5}, \sqrt{34}, 5 \)), so it can't be a rhombus or square.
  2. 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.