Questions: Given the points W(-4,3), X(2,6), L(4,2), and D(-2,-1), what is the most specific classification of the quadrilateral?

Solution Steps

To classify the quadrilateral formed by the given points, we need to calculate the lengths of all sides and the diagonals. By comparing these lengths, we can determine if the quadrilateral is a square, rectangle, rhombus, parallelogram, or trapezoid. Specifically, we will check for equal side lengths and parallel opposite sides.

To classify the quadrilateral, we first calculate the lengths of its sides using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• \( W(-4, 3) \) to \( I(2, 6) \):
  \[ WI = \sqrt{(2 - (-4))^2 + (6 - 3)^2} = \sqrt{6^2 + 3^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.708 \]

• \( I(2, 6) \) to \( L(4, 2) \):
  \[ IL = \sqrt{(4 - 2)^2 + (2 - 6)^2} = \sqrt{2^2 + (-4)^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.472 \]

• \( L(4, 2) \) to \( D(-2, -1) \):
  \[ LD = \sqrt{(-2 - 4)^2 + (-1 - 2)^2} = \sqrt{(-6)^2 + (-3)^2} = \sqrt{36 + 9} = \sqrt{45} \approx 6.708 \]

• \( D(-2, -1) \) to \( W(-4, 3) \):
  \[ DW = \sqrt{(-4 - (-2))^2 + (3 - (-1))^2} = \sqrt{(-2)^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} \approx 4.472 \]

Next, we calculate the lengths of the diagonals:

• \( W(-4, 3) \) to \( L(4, 2) \):
  \[ WL = \sqrt{(4 - (-4))^2 + (2 - 3)^2} = \sqrt{8^2 + (-1)^2} = \sqrt{64 + 1} = \sqrt{65} \approx 8.062 \]

• \( I(2, 6) \) to \( D(-2, -1) \):
  \[ ID = \sqrt{(-2 - 2)^2 + (-1 - 6)^2} = \sqrt{(-4)^2 + (-7)^2} = \sqrt{16 + 49} = \sqrt{65} \approx 8.062 \]

Based on the calculated side lengths and diagonals:

• Opposite sides are equal: \( WI = LD \approx 6.708 \) and \( IL = DW \approx 4.472 \).
• Diagonals are equal: \( WL = ID \approx 8.062 \).

Since the opposite sides are equal and the diagonals are equal, the quadrilateral is a rectangle.

Final Answer