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To determine whether the points \( A(3, 7), B(-2, -1), C(4, 1), \) and \( D(9, 9) \) form a parallelogram, we can use the property that in a parallelogram, opposite sides are parallel. This means their slopes should be equal.

### Step-by-step solution:

1. **Calculate the slope of \( AB \):**
   The slope of a line through points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   For \( A(3, 7) \) and \( B(-2, -1) \):
   \[
   m_{AB} = \frac{-1 - 7}{-2 - 3} = \frac{-8}{-5} = \frac{8}{5}
   \]

2. **Calculate the slope of \( CD \):**
   For \( C(4, 1) \) and \( D(9, 9) \):
   \[
   m_{CD} = \frac{9 - 1}{9 - 4} = \frac{8}{5}
   \]
   So, \( m_{AB} = m_{CD} = \frac{8}{5} \), which means \( AB \parallel CD \).

3. **Calculate the slope of \( BC \):**
   For \( B(-2, -1) \) and \( C(4, 1) \):
   \[
   m_{BC} = \frac{1 - (-1)}{4 - (-2)} = \frac{2}{6} = \frac{1}{3}
   \]

4. **Calculate the slope of \( AD \):**
   For \( A(3, 7) \) and \( D(9, 9) \):
   \[
   m_{AD} = \frac{9 - 7}{9 - 3} = \frac{2}{6} = \frac{1}{3}
   \]
   So, \( m_{BC} = m_{AD} = \frac{1}{3} \), which means \( BC \parallel AD \).

Since both pairs of opposite sides are parallel (\( AB \parallel CD \) and \( BC \parallel AD \)), the four points form a **parallelogram**.

Would you like any further details on this explanation?

### Related questions:
1. How do you calculate the distance between two points?
2. What are the properties of a parallelogram besides parallel sides?
3. How do you find the area of a parallelogram using coordinates?
4. How would this method change if the points formed a rectangle or square?
5. What is the difference between slope and distance when verifying geometric shapes?

### Tip:
When verifying if four points form a parallelogram, you can also use vector addition to check if the diagonals bisect each other.

Determine whether or not the points A(3, 7), B(-2, -1), C(4, 1), and D(9, 9) form a parallelogram.

Learn how to determine if points A(3, 7), B(-2, -1), C(4, 1), and D(9, 9) form a parallelogram by calculating the slopes of the lines. This problem involves key concepts from coordinate geometry and is suitable for high school students.

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