Given the set of points as vertices of quadrilateral QRST, determine whether it is a rhombus, a rectangle, or a square. Choose all that apply. Q(1, 2), R(-2, -1\( ), S(1,-4), T(4,-1) \) Select one or more: a. Kite b. square c. rhombus d. None of these e. Rectangle

To determine the type of quadrilateral QRST formed by the points \( Q(1, 2) \), \( R(-2, -1) \), \( S(1, -4) \), and \( T(4, -1) \), we will follow these steps: 1. **Calculate the lengths of the sides** to check if it is a rhombus or a square. 2. **Calculate the lengths of the diagonals** to check if it is a rectangle or a square. 3. **Check the properties of the angles** to confirm the type of quadrilateral. ### Step 1: Calculate the lengths of the sides The distance between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by the formula: \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Let's calculate the lengths of the sides \( QR \), \( RS \), \( ST \), and \( TQ \). - **Length of \( QR \)**: \[ d_{QR} = \sqrt{((-2) - 1)^2 + ((-1) - 2)^2} = \sqrt{(-3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \] - **Length of \( RS \)**: \[ d_{RS} = \sqrt{(1 - (-2))^2 + ((-4) - (-1))^2} = \sqrt{(3)^2 + (-3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \] - **Length of \( ST \)**: \[ d_{ST} = \sqrt{(4 - 1)^2 + ((-1) - (-4))^2} = \sqrt{(3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \] - **Length of \( TQ \)**: \[ d_{TQ} = \sqrt{(1 - 4)^2 + (2 - (-1))^2} = \sqrt{(-3)^2 + (3)^2} = \sqrt{9 + 9} = \sqrt{18} = 3\sqrt{2} \] ### Step 2: Calculate the lengths of the diagonals Now, let's calculate the lengths of the diagonals \( QS \) and \( RT \). - **Length of \( QS \)**: \[ d_{QS} = \sqrt{(1 - 1)^2 + (2 - (-4))^2} = \sqrt{(0)^2 + (6)^2} = \sqrt{36} = 6 \] - **Length of \( RT \)**: \[ d_{RT} = \sqrt{(4 - (-2))^2 + ((-1) - (-1))^2} = \sqrt{(6)^2 + (0)^2} = \sqrt{36} = 6 \] ### Step 3: Analyze the properties - All sides are equal: \( QR = RS = ST = TQ = 3\sqrt{2} \) (indicating it could be a rhombus or square). - The diagonals are equal: \( QS = RT = 6 \) (indicating it could be a rectangle or square). ### Conclusion Since all sides are equal and the diagonals are equal, the quadrilateral QRST is a **square**. It is also a **rhombus** because all sides are equal. Thus, the correct options are: - b. square - c. rhombus The final answer is: - **b. square** - **c. rhombus**