Question

Use slopes to prove this is a rectangle. What is the slope of side XY? What is the slope of WX? Are these sides parallel or perpendicular? Slope of XY = -2, slope of WX = 2, these lines are perpendicular Slope of XY = -2, slope of WX = 2, these lines are parallel Slope of XY = -1, slope of WX = 1, these lines are parallel Slope of XY = -1, slope of WX = 1, these lines are perpendicular

Answer 1

W is (-2,3) X is, (1,6) Y is, (3,4) Z is, (0,1)

Answer 2

Which option is it?

Answer 3

To determine whether the sides XY and WX are parallel or perpendicular using their slopes, we can use the following rules:

1. **Parallel Lines**: Two lines are parallel if they have the same slope.
2. **Perpendicular Lines**: Two lines are perpendicular if the product of their slopes is -1.

Now, let's analyze the given slopes:

- Slope of side XY = -2
- Slope of side WX = 2

Now, we can check if these lines are parallel or perpendicular:

1. **Checking for Parallelism**:
- The slopes are -2 and 2, which are not the same. Therefore, lines XY and WX are not parallel.

2. **Checking for Perpendicularity**:
- The product of the slopes is:
\[
(-2) \times (2) = -4
\]
Since the result is not -1, the lines are also not perpendicular.

Based on this analysis, we can conclude that the correct option is neither parallel nor perpendicular from the choices you've provided. However, if we were to only consider the closest option based on the slopes that suggests some form of relationship, the statements about perpendicularity would be relevant.

If the slopes were meant to be different (the options given may indicate some confusion in the slopes provided), please clarify or check the slopes again, as there's no valid conclusion based on the slopes of -2 and 2 that would lead to the conclusions given in the options.

Thus, with the given slopes, **XY and WX are neither parallel nor perpendicular**. Would you like to provide a revised set of slopes for clarity?

Answer 4

To determine whether the figure formed by the points W, X, Y, and Z is a rectangle, we will calculate the slopes of the sides WX, XY, YZ, and ZW, and check their relationships.

### Step 1: Find the coordinates
- \( W = (-2, 3) \)
- \( X = (1, 6) \)
- \( Y = (3, 4) \)
- \( Z = (0, 1) \)

### Step 2: Calculate the slopes of the sides

#### Slope of WX
The slope formula is given by:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
For \( WX \):
\[
\text{slope of WX} = \frac{6 - 3}{1 - (-2)} = \frac{3}{3} = 1
\]

#### Slope of XY
For \( XY \):
\[
\text{slope of XY} = \frac{4 - 6}{3 - 1} = \frac{-2}{2} = -1
\]

#### Slope of YZ
For \( YZ \):
\[
\text{slope of YZ} = \frac{1 - 4}{0 - 3} = \frac{-3}{-3} = 1
\]

#### Slope of ZW
For \( ZW \):
\[
\text{slope of ZW} = \frac{3 - 1}{-2 - 0} = \frac{2}{-2} = -1
\]

### Step 3: Analyze slopes
Now we have:
- Slope of WX = \( 1 \)
- Slope of XY = \( -1 \)
- Slope of YZ = \( 1 \)
- Slope of ZW = \( -1 \)

### Step 4: Determine relationships
- Lines WX and YZ have the same slope (1), which means they are parallel.
- Lines XY and ZW also have the same slope (-1), which means they are parallel.

### Step 5: Check perpendicularity
Now, check the relationships between:
- WX (slope = 1) and XY (slope = -1):
- The product of the slopes: \( 1 \times (-1) = -1 \), which means they are perpendicular.

- WX (slope = 1) and ZW (slope = -1):
- The product of the slopes: \( 1 \times (-1) = -1 \), which means they are perpendicular.

- XY (slope = -1) and YZ (slope = 1):
- The product of the slopes: \( (-1) \times 1 = -1 \), which means they are perpendicular.

### Conclusion
Since opposite sides WX and YZ are parallel, and XY and ZW are parallel, and all adjacent sides are perpendicular to each other, we can conclude that the figure formed by the points W, X, Y, and Z is indeed a rectangle.

Answer 5

Based on the slopes we calculated:

- Slope of \(WX = 1\)
- Slope of \(XY = -1\)
- Slope of \(YZ = 1\)
- Slope of \(ZW = -1\)

The relationships between the sides are:

- \(WX\) and \(YZ\) are parallel (both have a slope of \(1\)).
- \(XY\) and \(ZW\) are parallel (both have a slope of \(-1\)).
- \(WX\) is perpendicular to \(XY\) (product of slopes is \(-1\)).
- \(YZ\) is perpendicular to \(ZW\) (product of slopes is \(-1\)).
- \(XY\) is perpendicular to \(ZW\) (product of slopes is \(-1\)).

### Therefore, the correct conclusion is:
**Slope of XY = -1, slope of WX = 1, these lines are perpendicular.**

So the relevant option from the provided choices would be:

1. **Slope of XY = -1, slope of WX = 1, these lines are perpendicular.**

This confirms the figure is a rectangle, as it has opposite sides that are parallel and adjacent sides that are perpendicular.