Determine if segments ab and CD are parallel perpendicular or neither

Question

1. A(-1,3) and B(1,5) & C(2,3) D(-2,7)

Answer 1

To determine if segments AB and CD are parallel, perpendicular, or neither, we need to find the slopes of both segments.

1. Calculate the slope of segment AB:

The formula for the slope (m) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For points A(-1, 3) and B(1, 5):

\[ m_{AB} = \frac{5 - 3}{1 - (-1)} = \frac{2}{1 + 1} = \frac{2}{2} = 1 \]

2. Calculate the slope of segment CD:

For points C(2, 3) and D(-2, 7):

\[ m_{CD} = \frac{7 - 3}{-2 - 2} = \frac{4}{-4} = -1 \]

3. Determine the relationship between the segments:

• Two lines are parallel if their slopes are equal.
• Two lines are perpendicular if their slopes are negative reciprocals of each other (i.e., \(m_1 \cdot m_2 = -1\)).

Here, we have:

• \(m_{AB} = 1\)
• \(m_{CD} = -1\)

Since the product of the slopes is:

\[ m_{AB} \cdot m_{CD} = 1 \cdot (-1) = -1 \]

This shows that AB and CD are perpendicular.

Therefore, the answer is: perpendicular.

Answer 2

Determine if segments ab and CD are parallel perpendicular or neither
1. A(3,8) and B(-6,5) & C(-6,7) D(-3,8)

Answer 3

To determine if segments AB and CD are parallel, perpendicular, or neither, we'll need to calculate the slopes of both segments.

1. Calculate the slope of segment AB:

The formula for the slope \( m \) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

For points A(3, 8) and B(-6, 5):

\[ m_{AB} = \frac{5 - 8}{-6 - 3} = \frac{-3}{-9} = \frac{1}{3} \]

2. Calculate the slope of segment CD:

For points C(-6, 7) and D(-3, 8):

\[ m_{CD} = \frac{8 - 7}{-3 - (-6)} = \frac{1}{-3 + 6} = \frac{1}{3} \]

3. Determine the relationship between the segments:

• Two lines are parallel if their slopes are equal.
• Two lines are perpendicular if their slopes are negative reciprocals of each other (i.e., \( m_1 \cdot m_2 = -1 \)).

Here we have:

• \( m_{AB} = \frac{1}{3} \)
• \( m_{CD} = \frac{1}{3} \)

Since the slopes are equal (\( m_{AB} = m_{CD} \)), this indicates that the two segments are parallel.

Conclusion:

Segments AB and CD are parallel.

Answer 4

Determine if segments ab and CD are parallel perpendicular or neither
1. A(2,-7) and B(2,3) & C(-4,0) D(-4,-5)

Answer 5

To determine if segments AB and CD are parallel, perpendicular, or neither, we need to calculate the slopes of both segments.

Step 1: Calculate the slope of segment AB

For points A(2, -7) and B(2, 3):

Using the slope formula:

\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]

Substituting the coordinates of points A and B:

\[ m_{AB} = \frac{3 - (-7)}{2 - 2} = \frac{3 + 7}{0} = \frac{10}{0} \]

Since the denominator is zero, the slope \(m_{AB}\) is undefined. This means that segment AB is a vertical line.

Step 2: Calculate the slope of segment CD

For points C(-4, 0) and D(-4, -5):

Using the slope formula again:

\[ m_{CD} = \frac{-5 - 0}{-4 - (-4)} = \frac{-5}{-4 + 4} = \frac{-5}{0} \]

Again, since the denominator is zero, the slope \(m_{CD}\) is also undefined. This means that segment CD is also a vertical line.

Conclusion

Since both segments AB and CD have undefined slopes (both are vertical lines), they are parallel to each other.

Therefore, the answer is: parallel.

Answer 6

Given the following four lines, pick the true statement
Line one: 3y = 4x + 3
Line 2: 4y = 3x - 4
Line 3: 3x + 4y = 8
Line 4: x + 3y = -6

A. Lines 1 and 4 are parallel
B. Lines two and three are parallel
C. Lines 2 and 4 perpendicular
D. Lines 1 and 2 are perpendicular

Answer 7

To determine which statements are true regarding the four lines, we need to find the slopes of each line by rewriting them in slope-intercept form \(y = mx + b\), where \(m\) is the slope.

Line 1: \(3y = 4x + 3\)

Dividing by 3:

\[ y = \frac{4}{3}x + 1 \]

Slope of Line 1: \(m_1 = \frac{4}{3}\)

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Line 2: \(4y = 3x - 4\)

Dividing by 4:

\[ y = \frac{3}{4}x - 1 \]

Slope of Line 2: \(m_2 = \frac{3}{4}\)

---

Line 3: \(3x + 4y = 8\)

Rearranging to slope-intercept form:

\[ 4y = -3x + 8 \quad \Rightarrow \quad y = -\frac{3}{4}x + 2 \]

Slope of Line 3: \(m_3 = -\frac{3}{4}\)

---

Line 4: \(x + 3y = -6\)

Rearranging to slope-intercept form:

\[ 3y = -x - 6 \quad \Rightarrow \quad y = -\frac{1}{3}x - 2 \]

Slope of Line 4: \(m_4 = -\frac{1}{3}\)

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Now evaluate the statements:

A. Lines 1 and 4 are parallel.
\(m_1 = \frac{4}{3}\), \(m_4 = -\frac{1}{3}\)
Not parallel.

B. Lines 2 and 3 are parallel.
\(m_2 = \frac{3}{4}\), \(m_3 = -\frac{3}{4}\)
Not parallel.

C. Lines 2 and 4 are perpendicular.
\(m_2 = \frac{3}{4}\), \(m_4 = -\frac{1}{3}\)
To check if they are perpendicular, the product of their slopes should equal -1:
\(\frac{3}{4} \cdot -\frac{1}{3} = -\frac{1}{4}\) (not -1, so not perpendicular).

D. Lines 1 and 2 are perpendicular.
\(m_1 = \frac{4}{3}\), \(m_2 = \frac{3}{4}\)
To check if they are perpendicular:
\(\frac{4}{3} \cdot \frac{3}{4} = 1\) (not -1, so not perpendicular).

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Conclusion:

None of the statements (A, B, C, D) are true.

Answer 8

Are you sure? Because one of them has to be true

Answer 9

Let's go through the calculations for determining the relationships between the lines one more time to accurately identify any true statements.

Given Lines:

1. Line 1: \(3y = 4x + 3\)
2. Line 2: \(4y = 3x - 4\)
3. Line 3: \(3x + 4y = 8\)
4. Line 4: \(x + 3y = -6\)

Step 1: Convert to Slope-Intercept Form

Line 1: \(3y = 4x + 3\)

Dividing by 3:

\[ y = \frac{4}{3}x + 1 \]

Slope of Line 1: \(m_1 = \frac{4}{3}\)

---

Line 2: \(4y = 3x - 4\)

Dividing by 4:

\[ y = \frac{3}{4}x - 1 \]

Slope of Line 2: \(m_2 = \frac{3}{4}\)

---

Line 3: \(3x + 4y = 8\)

Rearranging:

\[ 4y = -3x + 8 \ \Rightarrow \ y = -\frac{3}{4}x + 2 \]

Slope of Line 3: \(m_3 = -\frac{3}{4}\)

---

Line 4: \(x + 3y = -6\)

Rearranging:

\[ 3y = -x - 6 \ \Rightarrow \ y = -\frac{1}{3}x - 2 \]

Slope of Line 4: \(m_4 = -\frac{1}{3}\)

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Step 2: Evaluate Each Statement

A. Lines 1 and 4 are parallel.

• \(m_1 = \frac{4}{3}, m_4 = -\frac{1}{3}\) → Not parallel (\(m_1 \neq m_4\))

B. Lines 2 and 3 are parallel.

• \(m_2 = \frac{3}{4}, m_3 = -\frac{3}{4}\) → Not parallel (\(m_2 \neq m_3\))

C. Lines 2 and 4 are perpendicular.

• Product of slopes: \(m_2 \cdot m_4 = \frac{3}{4} \cdot -\frac{1}{3} = -\frac{1}{4}\) → Not perpendicular (\(-1 \neq -\frac{1}{4}\))

D. Lines 1 and 2 are perpendicular.

• Product of slopes: \(m_1 \cdot m_2 = \frac{4}{3} \cdot \frac{3}{4} = 1\) → Not perpendicular (\(-1 \neq 1\))

Now let's reevaluate for parallel or perpendicular using the proper set of calculations.

Key Check of Perpendicularity:

Two lines are perpendicular if \(m_1 \cdot m_2 = -1\).

• First check if lines 2 and 3 are perpendicular:
  • \(m_2 = \frac{3}{4}\)
  • \(m_3 = -\frac{3}{4}\)

Recap Slopes for Perpendicular

• Checking lines 1 and 3:
  • If you multiply \( \frac{4}{3} by -\frac{3}{4} = -1\) confirms perpendicular.

Summary of Statements:

The true relations based on all checking shows that:

• Lines 1 and 3 are perpendicular, confirming what we may not have seen first.
• Thus confirming Lines 2 and 4 aren't, nor parallel.

True Affirmation: Lines 1 and 3 are perpendicular. Thank you for your patience; the best answer realizing true corrections means lines 1 and 3 have true perpendicular! To summarize the original options presented, none appears correct originally but if option modification allowed, Lines 1 and 3 would be stated!

Answer 10

I thought b was true

Answer 11

Let's carefully check the relationship between lines 2 and 3 again and clarify the definitions of parallelism and perpendicularity.

Recap of Line Equations:

1. Line 1: \(3y = 4x + 3\) → \(y = \frac{4}{3}x + 1\)
   Slope: \(m_1 = \frac{4}{3}\)

2. Line 2: \(4y = 3x - 4\) → \(y = \frac{3}{4}x - 1\)
   Slope: \(m_2 = \frac{3}{4}\)

3. Line 3: \(3x + 4y = 8\) → \(4y = -3x + 8\) → \(y = -\frac{3}{4}x + 2\)
   Slope: \(m_3 = -\frac{3}{4}\)

4. Line 4: \(x + 3y = -6\) → \(3y = -x - 6\) → \(y = -\frac{1}{3}x - 2\)
   Slope: \(m_4 = -\frac{1}{3}\)

Check for Parallelism and Perpendicularity:

Parallel Lines: Two lines are parallel if their slopes are equal.

Perpendicular Lines: Two lines are perpendicular if the product of their slopes is \(-1\).

Evaluating Statement B:

B. Lines 2 and 3 are parallel.

• Slope of Line 2: \(m_2 = \frac{3}{4}\)
• Slope of Line 3: \(m_3 = -\frac{3}{4}\)

Since:

\[ m_2 = \frac{3}{4} \quad \text{and} \quad m_3 = -\frac{3}{4} \]

The slopes are not equal; therefore, Lines 2 and 3 are not parallel.

Final Clarification of All Statements:

• A. Lines 1 and 4 are parallel: Not true, since the slopes differ.
• B. Lines 2 and 3 are parallel: Not true, as the slopes differ.
• C. Lines 2 and 4 are perpendicular: Checking gives: \[ m_2 \cdot m_4 = \frac{3}{4} \cdot -\frac{1}{3} = -\frac{1}{4} \] Not true, since the product is not -1.
• D. Lines 1 and 2 are perpendicular: Checking gives: \[ m_1 \cdot m_2 = \frac{4}{3} \cdot \frac{3}{4} = 1 \] Not true, as the product is not -1.

Conclusion:

None of the statements are true as provided. I apologize for the oversight, especially acknowledging your interest that "B" could have been correct but reconfirming does not support that.

However, a correct recap does show:

• Lines 1 and 3 are perpendicular, as \( \frac{4}{3} \cdot -\frac{3}{4} = -1 \).

Thank you for your patience; please feel free to ask any follow-up questions or if you'd like to explore another area!