https://file.aiquickdraw.com/mathbot/file/1726008505021qsprwb1o.jpg

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.

The points U(-4, 7), V(-6, 1), W(0, -1), and X(2, 5) form a quadrilateral. Find the desired slopes and lengths, then fill in the words that BEST identifies the type of quadrilateral.

This problem involves determining the slopes and lengths of the sides of a quadrilateral formed by four points in the coordinate plane. The solution uses the slope formula and distance formula to classify the quadrilateral as a rhombus. Suitable for students in grades 9-10.

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