https://file.aiquickdraw.com/mathbot/file/17270566099386qy9i8pu.jpg

To solve the problem, we need to determine the equation of a line that:

1. Is perpendicular to the green line.
2. Passes through the purple point.

The equation of a line in **slope-intercept form** is given by:

\[
y = mx + b
\]

where \(m\) is the slope and \(b\) is the y-intercept.

### Step 1: Find the slope of the green line
We can visually estimate the slope of the green line by selecting two points on the green line and using the slope formula:

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

From the graph:
- The green line passes through the point \((0, -10)\) and the point \((5, -5)\).
- Using these points, the slope \(m_{\text{green}}\) is:

\[
m_{\text{green}} = \frac{-5 - (-10)}{5 - 0} = \frac{5}{5} = 1
\]

So, the slope of the green line is \(m_{\text{green}} = 1\).

### Step 2: Find the slope of the perpendicular line
If two lines are perpendicular, their slopes are negative reciprocals of each other. Therefore, the slope \(m_{\text{perpendicular}}\) of the line perpendicular to the green line is:

\[
m_{\text{perpendicular}} = -\frac{1}{1} = -1
\]

### Step 3: Use the point-slope form
Now, we use the purple point to find the y-intercept of the perpendicular line. The coordinates of the purple point are \((20, 0)\). 

Using the slope-intercept form \(y = mx + b\) with \(m = -1\) and the point \((20, 0)\):

\[
0 = -1(20) + b
\]

Solving for \(b\):

\[
0 = -20 + b \implies b = 20
\]

Thus, the equation of the line is:

\[
y = -x + 20
\]

### Step 4: Compare with the answer choices
The correct answer is:

\[
y = -x + 20
\]

However, none of the provided options match this exactly, suggesting a potential discrepancy with the given choices or a small detail in their labeling. Based on the graph and analysis, the equation \(y = -x + 20\) is correct.

Let me know if you need more clarification or if you have any further questions! Here are 5 relative questions to consider:

1. How do you find the slope of a line given two points?
2. What is the relationship between the slopes of perpendicular lines?
3. How does point-slope form help in finding the equation of a line?
4. How do you convert point-slope form to slope-intercept form?
5. What would happen if the line passed through a different point on the graph?

**Tip:** When working with slopes, always check for negative reciprocal relationships when dealing with perpendicular lines.

What is the equation in slope-intercept form of the line that crosses through the purple point and is perpendicular to the green line?

Learn how to find the equation of a line perpendicular to a given line that passes through a specific point. This problem involves calculating the slope, applying perpendicular slope relationships, and using the slope-intercept form of a line equation. Suitable for students in Grades 8-10.

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