https://file.aiquickdraw.com/mathbot/file/1727083049497wrsfb555.jpg

The question asks for the equation of the line **perpendicular to the given line that contains the point \( P \)**.

Let's analyze and solve the problems one by one from the image.

### Problem 19:
- Point: \( P(6, 6) \)
- Equation of the given line: \( y = \frac{2}{3}x \)

#### Solution for Problem 19:
1. **Slope of the given line**: The slope \( m_1 \) of the given line \( y = \frac{2}{3}x \) is \( \frac{2}{3} \).
2. **Slope of the perpendicular line**: The slope \( m_2 \) of a line perpendicular to another is the negative reciprocal of the original slope. So, \( m_2 = -\frac{3}{2} \).
3. **Equation of the line**: Using the point-slope form of the line \( y - y_1 = m(x - x_1) \), with point \( P(6, 6) \) and slope \( m_2 = -\frac{3}{2} \):
   \[
   y - 6 = -\frac{3}{2}(x - 6)
   \]
4. **Simplify**:
   \[
   y - 6 = -\frac{3}{2}x + 9
   \]
   \[
   y = -\frac{3}{2}x + 15
   \]
   So, the equation of the perpendicular line is:
   \[
   y = -\frac{3}{2}x + 15
   \]

---

### Problem 20:
- Point: \( P(4, 0) \)
- Equation of the given line: \( y = \frac{1}{2}x - 5 \)

#### Solution for Problem 20:
1. **Slope of the given line**: The slope \( m_1 \) of the line \( y = \frac{1}{2}x - 5 \) is \( \frac{1}{2} \).
2. **Slope of the perpendicular line**: The slope of the perpendicular line will be the negative reciprocal, \( m_2 = -2 \).
3. **Equation of the line**: Using the point \( P(4, 0) \) and slope \( m_2 = -2 \):
   \[
   y - 0 = -2(x - 4)
   \]
4. **Simplify**:
   \[
   y = -2x + 8
   \]
   So, the equation of the perpendicular line is:
   \[
   y = -2x + 8
   \]

---

### Problem 21:
- Point: \( P(4, 4) \)
- Equation of the given line: \( y = -2x - 8 \)

#### Solution for Problem 21:
1. **Slope of the given line**: The slope \( m_1 \) of the line \( y = -2x - 8 \) is \( -2 \).
2. **Slope of the perpendicular line**: The slope of the perpendicular line will be the negative reciprocal, \( m_2 = \frac{1}{2} \).
3. **Equation of the line**: Using the point \( P(4, 4) \) and slope \( m_2 = \frac{1}{2} \):
   \[
   y - 4 = \frac{1}{2}(x - 4)
   \]
4. **Simplify**:
   \[
   y - 4 = \frac{1}{2}x - 2
   \]
   \[
   y = \frac{1}{2}x + 2
   \]
   So, the equation of the perpendicular line is:
   \[
   y = \frac{1}{2}x + 2
   \]

---

Would you like more details on any part of the solution? Here are some questions for further practice:

1. How do you find the slope of a perpendicular line given the original slope?
2. How can you verify that two lines are perpendicular using their slopes?
3. What is the point-slope form of a line, and when is it useful?
4. How can you convert point-slope form into slope-intercept form?
5. How would you approach this problem if the point didn't lie directly on the given line?

**Tip:** Always check that the new line is indeed perpendicular by verifying that the product of the slopes of two perpendicular lines equals -1.

Write an equation of the line perpendicular to the given line that contains the point P. Problem 19: P(6, 6), y = (2/3)x. Problem 20: P(4, 0), y = (1/2)x - 5. Problem 21: P(4, 4), y = -2x - 8.

Learn how to write the equation of a line perpendicular to a given line through a specific point, with examples involving slopes and point-slope form. Suitable for students in Grades 8-10.

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