Question

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

Answer 1

The rectangle is ABCD. A is, (0,0) B is, (4,-4) C is (7,-1) D is, (3,3)

Answer 2

Choose the answer

Answer 3

To prove that a quadrilateral is a rectangle using slopes, you need to show that either pairs of opposite sides are parallel or that adjacent sides are perpendicular.

1. **Finding the slopes:**
- The slope of a line is calculated using the formula \((y_2 - y_1) / (x_2 - x_1)\).
- From your provided slopes, we’ll assume that sides AD and DC correspond to the slopes given.

2. **Analyzing slopes:**
- If you state side AD has a slope of "X" and side DC has a slope of "Y," we will need those specific values to determine if they’re parallel or perpendicular.
- Two lines are parallel if their slopes are equal (\( m_1 = m_2 \)).
- Two lines are perpendicular if the product of their slopes equals \(-1\) (\( m_1 \times m_2 = -1 \)).

3. **Given information:**
- Unfortunately, you haven't provided the specific slopes for sides AD and DC. However, you provided possible responses that include pairs of slopes.

4. **Responses Analysis:**
- If we take one of your responses, let’s analyze:
- **Slope of AD = 2, Slope of DC = -2:**
- These slopes are perpendicular since \(2 \times -2 = -4\), so this is incorrect as they should equal \(-1\).
- **Slope of AD = 1, Slope of DC = -1:**
- These slopes are perpendicular since \(1 \times -1 = -1\).
- **Slope of AD = -2, Slope of DC = 2:**
- Again, these slopes are perpendicular since \(-2 \times 2 = -4\), so this is also incorrect.

5. **Conclusion:**
- To conclusively prove that the quadrilateral is a rectangle, you need to determine if both pairs of opposite sides are either parallel or if adjacent sides are perpendicular.

6. **Final Summary:**
- Please confirm the specified slopes for sides AD and DC, and I can help determine if they help prove that the shape is a rectangle based on being parallel or perpendicular.

If you provide the accurate slopes for sides AD and DC, I can provide explicit conclusions regarding the conditions necessary for proving if the quadrilateral is a rectangle!

Answer 4

Let's find the slopes of the sides of rectangle ABCD, with the given coordinates:

- \( A(0, 0) \)
- \( B(4, -4) \)
- \( C(7, -1) \)
- \( D(3, 3) \)

### Step 1: Determine the slopes of the sides

1. **Slope of AB:**
\[
\text{slope of AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-4 - 0}{4 - 0} = \frac{-4}{4} = -1
\]

2. **Slope of BC:**
\[
\text{slope of BC} = \frac{-1 - (-4)}{7 - 4} = \frac{3}{3} = 1
\]

3. **Slope of CD:**
\[
\text{slope of CD} = \frac{3 - (-1)}{3 - 7} = \frac{4}{-4} = -1
\]

4. **Slope of DA:**
\[
\text{slope of DA} = \frac{0 - 3}{0 - 3} = \frac{-3}{-3} = 1
\]

### Step 2: Analyze the slopes

- **AB and CD have the same slope:**
- \( \text{slope of AB} = -1 \)
- \( \text{slope of CD} = -1 \)

**Conclusion:** AB is parallel to CD.

- **BC and DA have the same slope:**
- \( \text{slope of BC} = 1 \)
- \( \text{slope of DA} = 1 \)

**Conclusion:** BC is parallel to DA.

### Step 3: Check perpendicularity of adjacent sides

- **Product of slopes of AB and BC:**
\[
\text{slope of AB} \times \text{slope of BC} = -1 \times 1 = -1
\]
**Conclusion:** AB is perpendicular to BC.

- **Product of slopes of BC and CD:**
\[
\text{slope of BC} \times \text{slope of CD} = 1 \times -1 = -1
\]
**Conclusion:** BC is perpendicular to CD.

- **Product of slopes of CD and DA:**
\[
\text{slope of CD} \times \text{slope of DA} = -1 \times 1 = -1
\]
**Conclusion:** CD is perpendicular to DA.

- **Product of slopes of DA and AB:**
\[
\text{slope of DA} \times \text{slope of AB} = 1 \times -1 = -1
\]
**Conclusion:** DA is perpendicular to AB.

### Final conclusion

Since we have:
- \( AB \parallel CD \) and \( BC \parallel DA \)
- All adjacent sides are perpendicular (AB ⊥ BC, BC ⊥ CD, CD ⊥ DA, DA ⊥ AB)

Thus, quadrilateral ABCD is a rectangle.

Answer 5

Based on the calculations provided, here is the answer that matches the situation:

- The slopes of AB and CD are the same, indicating they are parallel.
- The slopes of BC and DA are also the same, indicating they are parallel.
- All adjacent sides are perpendicular to each other.

Therefore, the final conclusion is:

**ABCD is a rectangle.**

If you need to choose an answer from the previous slope-related statements you provided, the one that correctly states the relationship between the slopes is:
1. **Slope of XY = 1, slope of WX = -1, these lines are perpendicular** (if XY corresponds to one side and WX to another from the rectangle).

If you have other specific choices or options in mind, please share them so I can help you select the most accurate one!