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.