Given the points A(-2, 3), B(6, 1), C(5,-2) and D(-3, 0), show that ABCD is a parallelogram.

To prove that the quadrilateral ABCD is a parallelogram, we need to show that the opposite sides are parallel. This can be done by proving that the slopes of opposite sides are equal. 

### Step 1: Find the slope of AB and CD.
The formula for the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]

#### Slope of AB:
The coordinates of A and B are \((-2, 3)\) and \((6, 1)\), respectively.

\[
\text{slope of AB} = \frac{1 - 3}{6 - (-2)} = \frac{-2}{6 + 2} = \frac{-2}{8} = -\frac{1}{4}
\]

#### Slope of CD:
The coordinates of C and D are \((5, -2)\) and \((-3, 0)\), respectively.

\[
\text{slope of CD} = \frac{0 - (-2)}{-3 - 5} = \frac{2}{-8} = -\frac{1}{4}
\]

Since the slopes of AB and CD are equal \((-1/4)\), the sides AB and CD are parallel.

### Step 2: Find the slope of AD and BC.

#### Slope of AD:
The coordinates of A and D are \((-2, 3)\) and \((-3, 0)\), respectively.

\[
\text{slope of AD} = \frac{0 - 3}{-3 - (-2)} = \frac{-3}{-3 + 2} = \frac{-3}{-1} = 3
\]

#### Slope of BC:
The coordinates of B and C are \((6, 1)\) and \((5, -2)\), respectively.

\[
\text{slope of BC} = \frac{-2 - 1}{5 - 6} = \frac{-3}{-1} = 3
\]

Since the slopes of AD and BC are equal (3), the sides AD and BC are parallel.

### Conclusion:
Since opposite sides AB and CD, as well as AD and BC, are parallel, quadrilateral ABCD is a parallelogram.

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Would you like more details or have any questions? Here are some related questions to explore:

1. How do you determine if a quadrilateral is a rectangle?
2. How do you calculate the area of a parallelogram given four points?
3. What are the properties of a parallelogram?
4. How do you find the lengths of the sides of a parallelogram using coordinates?
5. Can a parallelogram be inscribed in a circle?

**Tip:** When checking for parallel lines in coordinate geometry, always compare the slopes of opposite sides.

Given the points A(-2, 3), B(6, 1), C(5, -2), and D(-3, 0), show that ABCD is a parallelogram.

Learn how to prove that a quadrilateral ABCD is a parallelogram using coordinate geometry. This step-by-step solution involves calculating slopes of opposite sides to confirm they are parallel. Suitable for students in Grades 9-10.

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