The quadrilateral with vertices P(-3, 2), Q(-5,-5), R(2, -3) and S (4,4) is a 

Options

A. rectangle
B. square
C. rhombus
D. kite

Answer

To determine the type of quadrilateral formed by the vertices P(-3, 2), Q(-5, -5), R(2, -3), and S(4, 4), we will follow these steps:

### Step 1: Plot the Points
First, we will plot the points on a coordinate plane:
- P(-3, 2) is in the second quadrant.
- Q(-5, -5) is in the third quadrant.
- R(2, -3) is in the fourth quadrant.
- S(4, 4) is in the first quadrant.

### Step 2: Calculate the Lengths of the Sides
We will use the distance formula to calculate the lengths of the sides PQ, QR, RS, and SP.

The distance formula is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

#### Length of PQ:
\[ PQ = \sqrt{(-5 - (-3))^2 + (-5 - 2)^2} = \sqrt{(-5 + 3)^2 + (-5 - 2)^2} = \sqrt{(-2)^2 + (-7)^2} = \sqrt{4 + 49} = \sqrt{53} \]

#### Length of QR:
\[ QR = \sqrt{(2 - (-5))^2 + (-3 - (-5))^2} = \sqrt{(2 + 5)^2 + (-3 + 5)^2} = \sqrt{(7)^2 + (2)^2} = \sqrt{49 + 4} = \sqrt{53} \]

#### Length of RS:
\[ RS = \sqrt{(4 - 2)^2 + (4 - (-3))^2} = \sqrt{(4 - 2)^2 + (4 + 3)^2} = \sqrt{(2)^2 + (7)^2} = \sqrt{4 + 49} = \sqrt{53} \]

#### Length of SP:
\[ SP = \sqrt{(-3 - 4)^2 + (2 - 4)^2} = \sqrt{(-7)^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53} \]

### Step 3: Compare the Lengths of the Sides
We find that:
- PQ = QR = RS = SP = \(\sqrt{53}\)

### Step 4: Check the Diagonals
Next, we will calculate the lengths of the diagonals PR and QS.

#### Length of PR:
\[ PR = \sqrt{(2 - (-3))^2 + (-3 - 2)^2} = \sqrt{(2 + 3)^2 + (-3 - 2)^2} = \sqrt{(5)^2 + (-5)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \]

#### Length of QS:
\[ QS = \sqrt{(4 - (-5))^2 + (4 - (-5))^2} = \sqrt{(4 + 5)^2 + (4 + 5)^2} = \sqrt{(9)^2 + (9)^2} = \sqrt{81 + 81} = \sqrt{162} = 9\sqrt{2} \]

### Step 5: Determine the Type of Quadrilateral
Since all sides are equal (PQ = QR = RS = SP) and the diagonals are not equal (PR ≠ QS), the quadrilateral is a **rhombus**.

### Conclusion
The quadrilateral with vertices P(-3, 2), Q(-5, -5), R(2, -3), and S(4, 4) is a **rhombus**.

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