https://file.aiquickdraw.com/mathbot/file/172809136588920m07rxq.jpg

Let's break down the problem to find the equation that is perpendicular to the graphed line and passes through the point \( (6, 2) \).

1. **Step 1: Determine the slope of the graphed line**
   From the image, the graphed line appears to be sloping downward. We need to calculate the slope, which can be done by identifying two points on the line and using the slope formula:
   \[
   m = \frac{{y_2 - y_1}}{{x_2 - x_1}}
   \]
   After approximating two points on the line, I estimate the slope to be \( m = -\frac{3}{4} \).

2. **Step 2: Find the slope of the perpendicular line**
   For two lines to be perpendicular, their slopes must be negative reciprocals of each other. If the slope of the original line is \( m = -\frac{3}{4} \), the slope of the perpendicular line would be:
   \[
   m_{\perp} = \frac{4}{3}
   \]

3. **Step 3: Use the point-slope form to find the equation**
   Now that we know the slope of the perpendicular line is \( \frac{4}{3} \) and it passes through the point \( (6, 2) \), we can use the point-slope form of a line equation:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( m = \frac{4}{3} \), \( x_1 = 6 \), and \( y_1 = 2 \):
   \[
   y - 2 = \frac{4}{3}(x - 6)
   \]
   Simplifying this equation:
   \[
   y - 2 = \frac{4}{3}x - 8
   \]
   \[
   y = \frac{4}{3}x - 6
   \]

4. **Step 4: Match with the given options**
   The correct equation is \( y = \frac{4}{3}x - 6 \), which matches the second option.

**Answer: Option 2: \( y = \frac{4}{3}x - 6 \)**

Would you like more details, or do you have any questions?

Here are 5 related questions to consider:
1. What is the general form of a line equation, and how can it be converted to slope-intercept form?
2. How do you find the slope between two points on a graph?
3. What is the relationship between the slopes of perpendicular lines?
4. How would you verify if two lines are perpendicular using their equations?
5. How can the point-slope form of a line equation help in solving geometry problems?

**Tip:** Always remember that perpendicular lines have slopes that are negative reciprocals of each other!

Which equation is perpendicular to the graphed line and goes through the point (6, 2)?

Learn how to find the equation of a line that is perpendicular to a given line and passes through the point (6, 2). The problem includes concepts of slopes, linear equations, and the relationship between perpendicular lines. Suitable for Grades 8-10.

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