Let's work through this problem step-by-step.

Points provided:

• $U(-4, 7)$
• $V(-6, 1)$
• $W(0, -1)$
• $X(2, 5)$

1. Slope Formula:

The slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is calculated using: $\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}$

2. Distance Formula:

The length (distance) between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by: $\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Now, I'll calculate the slopes and lengths for each pair of points.

(a) Slope and Length of $\overline{UV}$:

Points: $U(-4, 7)$, $V(-6, 1)$

• Slope of $\overline{UV}$: $\text{slope of } \overline{UV} = \frac{1 - 7}{-6 - (-4)} = \frac{-6}{-2} = 3$

• Length of $\overline{UV}$: $\text{length of } \overline{UV} = \sqrt{(-6 - (-4))^2 + (1 - 7)^2} = \sqrt{(-2)^2 + (-6)^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32$

(b) Slope and Length of $\overline{VW}$:

Points: $V(-6, 1)$, $W(0, -1)$

• Slope of $\overline{VW}$: $\text{slope of } \overline{VW} = \frac{-1 - 1}{0 - (-6)} = \frac{-2}{6} = -\frac{1}{3}$

• Length of $\overline{VW}$: $\text{length of } \overline{VW} = \sqrt{(0 - (-6))^2 + (-1 - 1)^2} = \sqrt{6^2 + (-2)^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.32$

(c) Slope and Length of $\overline{WX}$:

Points: $W(0, -1)$, $X(2, 5)$

• Slope of $\overline{WX}$: $\text{slope of } \overline{WX} = \frac{5 - (-1)}{2 - 0} = \frac{6}{2} = 3$

• Length of $\overline{WX}$: $\text{length of } \overline{WX} = \sqrt{(2 - 0)^2 + (5 - (-1))^2} = \sqrt{2^2 + 6^2} = \sqrt{4 + 36} = \sqrt{40} \approx 6.32$

(d) Slope and Length of $\overline{XU}$:

Points: $X(2, 5)$, $U(-4, 7)$

• Slope of $\overline{XU}$: $\text{slope of } \overline{XU} = \frac{7 - 5}{-4 - 2} = \frac{2}{-6} = -\frac{1}{3}$

• Length of $\overline{XU}$: $\text{length of } \overline{XU} = \sqrt{(-4 - 2)^2 + (7 - 5)^2} = \sqrt{(-6)^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} \approx 6.32$

3. Identifying the Type of Quadrilateral:

• Slopes of opposite sides $UV$ and $WX$ are both 3, indicating that these sides are parallel.
• Slopes of opposite sides $VW$ and $XU$ are both $-\frac{1}{3}$, so these sides are also parallel.
• All side lengths are equal $\approx 6.32$.

Thus, quadrilateral UVWX is a rhombus because it has all sides of equal length and opposite sides are parallel.

Summary:

• Slope of $UV$: $3$, Length: $\approx 6.32$
• Slope of $VW$: $-\frac{1}{3}$, Length: $\approx 6.32$
• Slope of $WX$: $3$, Length: $\approx 6.32$
• Slope of $XU$: $-\frac{1}{3}$, Length: $\approx 6.32$

The quadrilateral UVWX is a rhombus.

Would you like further clarification or details on any part? Here are 5 related questions for further practice:

1. How do you determine if a quadrilateral is a square instead of a rhombus?
2. What are the properties of a rectangle in terms of slopes and side lengths?
3. How can we verify if a given quadrilateral is a parallelogram using slopes?
4. If the side lengths were different, how could you identify the quadrilateral as a trapezoid?
5. What formula would you use to calculate the area of a rhombus given its diagonals?

Tip: Parallel lines always have equal slopes. Use this to check if sides of a quadrilateral are parallel.