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

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 2

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

Answer 3

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 4

Choose the answer

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!