To find the equation, in point-slope form, of the line that is perpendicular to a given line and passes through a specific point, we need to follow these steps:
1. Identify the slope of the given line.
2. Determine the negative reciprocal of that slope to find the slope of the perpendicular line.
3. Write the equation of the line passing through the given point using the point-slope form.
Let’s start by analyzing the given options.
### Step 1: Identify the slope of the given line
We are provided with four potential lines:
1. [tex]\( y + 5 = x + 2 \)[/tex]
2. [tex]\( y - 2 = x - 5 \)[/tex]
3. [tex]\( y - 5 = -(x - 2) \)[/tex]
4. [tex]\( y + 2 = -(x + 5) \)[/tex]
Let’s convert each of these lines into slope-intercept form [tex]\( y = mx + b \)[/tex].
1. [tex]\( y + 5 = x + 2 \)[/tex]
[tex]\[
y = x - 3
\][/tex]
The slope [tex]\( m \)[/tex] is 1.
2. [tex]\( y - 2 = x - 5 \)[/tex]
[tex]\[
y = x - 3
\][/tex]
The slope [tex]\( m \)[/tex] is 1.
3. [tex]\( y - 5 = -(x - 2) \)[/tex]
[tex]\[
y - 5 = -x + 2
\][/tex]
[tex]\[
y = -x + 7
\][/tex]
The slope [tex]\( m \)[/tex] is -1.
4. [tex]\( y + 2 = -(x + 5) \)[/tex]
[tex]\[
y + 2 = -x - 5
\][/tex]
[tex]\[
y = -x - 7
\][/tex]
The slope [tex]\( m \)[/tex] is -1.
### Step 2: Determine the slope of the perpendicular line
To find the line that is perpendicular, we need to find the negative reciprocal of the slope.
- For slopes of 1 (first two lines), the negative reciprocal is:
[tex]\[
-\frac{1}{1} = -1
\][/tex]
- For slopes of -1 (last two lines), the negative reciprocal is:
[tex]\[
-\frac{-1}{1} = 1
\][/tex]
### Step 3: Write the equation in point-slope form
We are asked for the equation of the line that is perpendicular to one of the given lines and passes through the point [tex]\( (2, 5) \)[/tex].
Given the different slopes we've identified, we find the line [tex]\( y - 5 = -(x - 2) \)[/tex] (third line from the choices) to be perpendicular to lines that have slopes of 1.
The point-slope form equation of the line passing through [tex]\( (2, 5) \)[/tex] with slope 1 (the negative reciprocal of -1) is:
[tex]\[
y - 5 = 1(x - 2)
\][/tex]
Therefore, the correct answer corresponds to the third option provided.
[tex]\[
\boxed{3}
\][/tex]
