2. DOROINATE GEOMETRY Graph each quadrilateral with the given vertice Determine whether the figure is a parallelogram. Justify your answer wi the method indicated. 24. $$ A(-3,4), B(4,5), C(5,-1), D(-2,-2) $$; Slope Formula

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To determine whether the quadrilateral $$ ABCD $$ with vertices $$ A(-3, 4) $$, $$ B(4, 5) $$, $$ C(5, -1) $$, and $$ D(-2, -2) $$ is a parallelogram, we'll follow these steps:

1. **Graph the Quadrilateral:**
   
   While I can't provide a visual graph here, you can plot the points on a coordinate plane:

- **A(-3, 4):** 3 units left of the origin and 4 units up.
   - **B(4, 5):** 4 units right of the origin and 5 units up.
   - **C(5, -1):** 5 units right of the origin and 1 unit down.
   - **D(-2, -2):** 2 units left of the origin and 2 units down.

Connect the points in the order $$ A \rightarrow B \rightarrow C \rightarrow D \rightarrow A $$ to form the quadrilateral.

2. **Determine if $$ ABCD $$ is a Parallelogram Using the Slope Formula:**

A quadrilateral is a parallelogram if both pairs of opposite sides are parallel. To check for parallelism, we'll calculate the slopes of opposite sides using the slope formula:

$$
   \text{Slope} = \frac{y_2 - y_1}{x_2 - x_1}
   $$

- **Slope of $$ AB $$:**
     $$
     m_{AB} = \frac{5 - 4}{4 - (-3)} = \frac{1}{7}
     $$

- **Slope of $$ CD $$:**
     $$
     m_{CD} = \frac{-2 - (-1)}{-2 - 5} = \frac{-1}{-7} = \frac{1}{7}
     $$

Since $$ m_{AB} = m_{CD} $$, sides $$ AB $$ and $$ CD $$ are parallel.

- **Slope of $$ AD $$:**
     $$
     m_{AD} = \frac{-2 - 4}{-2 - (-3)} = \frac{-6}{1} = -6
     $$

- **Slope of $$ BC $$:**
     $$
     m_{BC} = \frac{-1 - 5}{5 - 4} = \frac{-6}{1} = -6
     $$

Since $$ m_{AD} = m_{BC} $$, sides $$ AD $$ and $$ BC $$ are parallel.

3. **Conclusion:**

Both pairs of opposite sides ($$ AB $$ & $$ CD $$, and $$ AD $$ & $$ BC $$) are parallel because they have equal slopes. Therefore, the quadrilateral $$ ABCD $$ is a **parallelogram**.
The quadrilateral $$ ABCD $$ with vertices $$ A(-3,4) $$, $$ B(4,5) $$, $$ C(5,-1) $$, and $$ D(-2,-2) $$ is a parallelogram because both pairs of opposite sides are parallel.

2. DOROINATE GEOMETRY Graph each quadrilateral with the given vertice Determine whether the figure is a parallelogram. Justify your answer wi the method indicated. 24. \( A(-3,4), B(4,5), C(5,-1), D(-2,-2) \); Slope Formula

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