Question

Determine if segments ab and CD are parallel perpendicular or neither

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

Answer 1

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 2

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 3

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 4

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

Answer 5

I thought b was true

Answer 6

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 7

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 8

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 9

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**: \(m_1 = \frac{4}{3}\)

---

### 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**: \(m_4 = -\frac{1}{3}\)

---

### 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).

---

### Conclusion:

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

Answer 10

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**: \(m_4 = -\frac{1}{3}\)

---

### 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 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!