Let's solve the questions based on the points provided in the image: $A(-3, -2)$, $B(2, -2)$, $C(5, 2)$, and $D(0, 2)$.

Step 1: Check if ABCD is a parallelogram

A quadrilateral is a parallelogram if opposite sides are parallel. This requires checking slopes of opposite sides:

1. Slope of AB: $\text{slope}_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-2 - (-2)}{2 - (-3)} = \frac{0}{5} = 0$

2. Slope of CD: $\text{slope}_{CD} = \frac{2 - 2}{0 - 5} = \frac{0}{-5} = 0$ Since $\text{slope}_{AB} = \text{slope}_{CD}$, $AB \parallel CD$.

3. Slope of BC: $\text{slope}_{BC} = \frac{2 - (-2)}{5 - 2} = \frac{4}{3}$

4. Slope of AD: $\text{slope}_{AD} = \frac{2 - (-2)}{0 - (-3)} = \frac{4}{3}$ Since $\text{slope}_{BC} = \text{slope}_{AD}$, $BC \parallel AD$.

Since opposite sides are parallel, ABCD is a parallelogram.

Step 2: Check if ABCD is a rectangle

A parallelogram is a rectangle if adjacent sides are perpendicular. Check the product of the slopes of $AB$ and $BC$:

1. Product of slopes $AB$ and $BC$: $\text{slope}_{AB} \cdot \text{slope}_{BC} = 0 \cdot \frac{4}{3} = 0$ Since the product is $0$, $AB \perp BC$.

2. Similarly, $BC \perp CD$, $CD \perp AD$, and $AD \perp AB$ (can be verified).

Thus, ABCD is a rectangle.

Step 3: Check if ABCD is a rhombus

A parallelogram is a rhombus if all sides are equal. Use the distance formula to calculate the lengths of all sides:

1. Length of AB: $AB = \sqrt{(2 - (-3))^2 + (-2 - (-2))^2} = \sqrt{5^2 + 0^2} = 5$

2. Length of BC: $BC = \sqrt{(5 - 2)^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$

3. Length of CD: $CD = \sqrt{(5 - 0)^2 + (2 - 2)^2} = \sqrt{5^2 + 0^2} = 5$

4. Length of AD: $AD = \sqrt{(0 - (-3))^2 + (2 - (-2))^2} = \sqrt{3^2 + 4^2} = 5$

Since all sides are equal, ABCD is a rhombus.

Step 4: Check if ABCD is a square

A quadrilateral is a square if it is both a rectangle and a rhombus. Since $ABCD$ is both a rectangle and a rhombus, ABCD is a square.

Summary:

1. Is ABCD a parallelogram? Yes.
2. Is ABCD a rectangle? Yes.
3. Is ABCD a rhombus? Yes.
4. Is ABCD a square? Yes.

Would you like more details about any step or further clarification?

Related Questions:

1. How do you find slopes of lines to verify parallelism?
2. What makes a parallelogram a rectangle?
3. How does the distance formula help confirm congruent sides?
4. Can a shape be both a rectangle and a rhombus but not a square?
5. What conditions define a square in a coordinate plane?

Tip:

When checking for perpendicularity, always multiply the slopes of two lines. If the result is $-1$ or $0$, they are perpendicular!