To determine the type of quadrilateral formed by points $N(-9, 3), O(-8, -5), P(-1, -9), Q(-2, -1)$, we need to calculate the slopes and lengths of its sides.

1. Slope Calculation:

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 NO (between points $N(-9, 3)$ and $O(-8, -5)$):

$\text{slope of NO} = \frac{-5 - 3}{-8 - (-9)} = \frac{-8}{1} = -8$

• Slope of OP (between points $O(-8, -5)$ and $P(-1, -9)$):

$\text{slope of OP} = \frac{-9 - (-5)}{-1 - (-8)} = \frac{-9 + 5}{-1 + 8} = \frac{-4}{7}$

• Slope of PQ (between points $P(-1, -9)$ and $Q(-2, -1)$):

$\text{slope of PQ} = \frac{-1 - (-9)}{-2 - (-1)} = \frac{-1 + 9}{-2 + 1} = \frac{8}{-1} = -8$

• Slope of QN (between points $Q(-2, -1)$ and $N(-9, 3)$):

$\text{slope of QN} = \frac{3 - (-1)}{-9 - (-2)} = \frac{3 + 1}{-9 + 2} = \frac{4}{-7} = -\frac{4}{7}$

2. Length Calculation:

The formula for the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$\text{length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

• Length of NO:

$\text{length of NO} = \sqrt{(-8 - (-9))^2 + (-5 - 3)^2} = \sqrt{(1)^2 + (-8)^2} = \sqrt{1 + 64} = \sqrt{65}$

• Length of OP:

$\text{length of OP} = \sqrt{(-1 - (-8))^2 + (-9 - (-5))^2} = \sqrt{(7)^2 + (-4)^2} = \sqrt{49 + 16} = \sqrt{65}$

• Length of PQ:

$\text{length of PQ} = \sqrt{(-2 - (-1))^2 + (-1 - (-9))^2} = \sqrt{(-1)^2 + (8)^2} = \sqrt{1 + 64} = \sqrt{65}$

• Length of QN:

$\text{length of QN} = \sqrt{(-9 - (-2))^2 + (3 - (-1))^2} = \sqrt{(-7)^2 + (4)^2} = \sqrt{49 + 16} = \sqrt{65}$

3. Conclusion:

• The slopes of opposite sides $NO$ and $PQ$ are both $-8$, which means these sides are parallel.
• The slopes of opposite sides $OP$ and $QN$ are both $-\frac{4}{7}$, meaning these sides are also parallel.
• The lengths of all four sides are equal, $\sqrt{65}$.

Since both pairs of opposite sides are parallel and all sides have equal length, Quadrilateral NOPQ is a rhombus.

Summary of Answers:

• Slope of NO = $-8$
• Length of NO = $\sqrt{65}$
• Slope of OP = $-\frac{4}{7}$
• Length of OP = $\sqrt{65}$
• Slope of PQ = $-8$
• Length of PQ = $\sqrt{65}$
• Slope of QN = $-\frac{4}{7}$
• Length of QN = $\sqrt{65}$
• Quadrilateral NOPQ can BEST be described as a rhombus.

Would you like further details or clarification on any step?

Here are five related questions:

1. How do you determine if two sides of a quadrilateral are parallel?
2. What is the difference between a rhombus and a square?
3. Can a rhombus have right angles? If so, under what conditions?
4. What is the general formula for the area of a rhombus?
5. How does the Pythagorean theorem apply to calculating distances in the coordinate plane?

Tip: Remember, in any quadrilateral, the slopes of parallel sides will always be equal!