To solve the problem of finding the equation of a line that is perpendicular to a given line and passes through the point (5, 0), we need to take several steps:
### Step 1: Understanding Line Equations
To find the equation of the line that Leo drew, we need to understand the concept of perpendicular lines. Perpendicular lines have slopes whose product is -1. This means if we know the slope of the given line, we can determine the slope of Leo's perpendicular line.
### Step 2: Identifying the Slope of Perpendicular Line
1. To calculate the slope of a perpendicular line, we use the property:
[tex]\[
m_{\text{perpendicular}} = -\frac{1}{m_{\text{original}}}
\][/tex]
where [tex]\( m_{\text{original}} \)[/tex] is the slope of the original line.
### Step 3: Using the Point-Slope Form
2. With the perpendicular slope and the point (5, 0), we can use the point-slope form of a line equation:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
where [tex]\( (x_1, y_1) \)[/tex] is the point the line passes through and [tex]\(m\)[/tex] is the slope of the perpendicular line.
### Step 4: Verifying the Correct Option
Given the choices:
1. [tex]\( y - 0 = -2(x - 6) \)[/tex]
2. [tex]\( y + 5 = 2(x + 9) \)[/tex]
3. [tex]\( y + 5 = \frac{1}{2}(x + 6) \)[/tex]
4. [tex]\( y = 0 = \frac{1}{2} a - n \)[/tex]
We start by transforming or evaluating each to compare with our derived line equation form.
### Step 5: Checking the Options
1. Option 1:
[tex]\[
y - 0 = -2(x - 6) \implies y = -2(x - 6) \implies y = -2x + 12
\][/tex]
This line passes through [tex]\( (6, 0) \)[/tex], but the slope does not conform to the perpendicular requirement based on not knowing [tex]\( m_{\text{original}} \)[/tex].
2. Option 2:
[tex]\[
y + 5 = 2(x + 9) \implies y + 5 = 2x + 18 \implies y = 2x + 18 - 5 \implies y = 2x + 13
\][/tex]
This does not pass through (5, 0).
3. Option 3:
[tex]\[
y + 5 = \frac{1}{2}(x + 6) \implies y + 5 = \frac{1}{2} x + 3 \implies y = \frac{1}{2} x + 3 - 5 \implies y = \frac{1}{2} x - 2
\][/tex]
This also does not pass through (5, 0).
4. Option 4:
The format [tex]\(y = 0 = \frac{1}{2} a - n\)[/tex] is unclear and is not a standard linear equation form.
### Conclusion:
Since none of the given options seem to meet the criteria for perpendicularity properly or the point passage confirmation based on the lack of an original line’s slope, the problem cannot deduce a specific correct line equation from given choices confidently.
Therefore, the correct answer to choose here aligned with the given options and solution constraints is:
```
None
```
This suggests that the correct line equation is not easily derived from the given options without proper original line information.
Correct selection should be revisited with known original line specifics for precise perpendicular determination.
