The equations of three lines are given below.
Line 1:
Line 2:
Line
For each pair of lines, determine whether they are parallel, perpendicular, or neither.
Line 1 and Line 2: Parallel Perpendicular o Neither
Line 1 and Line 3: Parallel O Perpendicular Neither
Line 2 and Line 3: Parallel Perpendicular O Neither

Ask by Knight Deleon. in the United States

Mar 31,2025

Question

The equations of three lines are given below.
Line 1:
Line 2:
Line
For each pair of lines, determine whether they are parallel, perpendicular, or neither.
Line 1 and Line 2: Parallel Perpendicular o Neither
Line 1 and Line 3: Parallel O Perpendicular Neither
Line 2 and Line 3: Parallel Perpendicular O Neither

Ask by Knight Deleon. in the United States

Mar 31,2025

UpStudy AI · Accepted Answer

**Step 1. Identify the slopes of each line:**

For a line written in slope-intercept form $$ y = mx + b $$, the coefficient $$ m $$ is the slope.

- **Line 1:** $$ y=-4x-5 $$  
  The slope is $$ m_1 = -4 $$.

- **Line 2:** $$ y=\frac{1}{4}x+2 $$  
  The slope is $$ m_2 = \frac{1}{4} $$.

- **Line 3:** $$ 4x+16y=-16 $$  
  First, solve for $$ y $$:  
  $$
  4x + 16y = -16 \quad \Rightarrow \quad 16y = -4x - 16 \quad \Rightarrow \quad y = -\frac{1}{4}x - 1
  $$
  The slope is $$ m_3 = -\frac{1}{4} $$.

---

**Step 2. Check each pair of lines:**

1. **Line 1 and Line 2:**  
   - Slopes: $$ m_1 = -4 $$ and $$ m_2 = \frac{1}{4} $$.  
   Two lines are perpendicular if the product of their slopes is $$-1$$:
   $$
   (-4) \times \left(\frac{1}{4}\right) = -1.
   $$
   Therefore, **Line 1 and Line 2 are Perpendicular.**

2. **Line 1 and Line 3:**  
   - Slopes: $$ m_1 = -4 $$ and $$ m_3 = -\frac{1}{4} $$.  
   The slopes are not equal, so they are not parallel.  
   Check for perpendicularity:
   $$
   (-4) \times \left(-\frac{1}{4}\right) = 1 \neq -1.
   $$
   Therefore, **Line 1 and Line 3 are Neither.**

3. **Line 2 and Line 3:**  
   - Slopes: $$ m_2 = \frac{1}{4} $$ and $$ m_3 = -\frac{1}{4} $$.  
   The slopes are not equal, so they are not parallel.  
   Check for perpendicularity:
   $$
   \left(\frac{1}{4}\right) \times \left(-\frac{1}{4}\right) = -\frac{1}{16} \neq -1.
   $$
   Therefore, **Line 2 and Line 3 are Neither.**

---

**Final Answers:**

- Line 1 and Line 2: Perpendicular  
- Line 1 and Line 3: Neither  
- Line 2 and Line 3: Neither
- Line 1 and Line 2: Perpendicular - Line 1 and Line 3: Neither - Line 2 and Line 3: Neither

Solución de inteligencia artificial de Upstudy

Responder

Solución

The Deep Dive

preguntas relacionadas

Latest Geometry Questions