To determine the equation of a line that is perpendicular to a given line and passes through a specific point, follow these steps:
1. Find the slope of the given line.
The given line is [tex]\( y - 2 = x - 5 \)[/tex]. First, we need to rewrite this in slope-intercept form [tex]\( y = mx + b \)[/tex] where [tex]\( m \)[/tex] is the slope.
[tex]\[
y - 2 = x - 5
\][/tex]
Add 2 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = x - 3
\][/tex]
From [tex]\( y = x - 3 \)[/tex], we see that the slope ([tex]\( m \)[/tex]) of the given line is 1.
2. Determine the slope of the perpendicular line.
Perpendicular lines have slopes that are negative reciprocals of each other. So, if the slope of the given line is 1, the slope of the perpendicular line ([tex]\( m_{\text{perp}} \)[/tex]) will be:
[tex]\[
m_{\text{perp}} = -\frac{1}{1} = -1
\][/tex]
3. Use the point-slope form to write the equation of the perpendicular line.
The point-slope form of a line's equation is:
[tex]\[
y - y_1 = m(x - x_1)
\][/tex]
Here, the point ([tex]\( x_1, y_1 \)[/tex]) is (2, 5) and the slope ([tex]\( m \)[/tex]) is -1. Substitute these values into the equation:
[tex]\[
y - 5 = -1(x - 2)
\][/tex]
4. Verify and simplify if needed.
Let's simplify the equation to confirm its form:
[tex]\[
y - 5 = -1(x - 2)
\][/tex]
The simplified form of this equation is:
[tex]\[
y - 5 = -x + 2
\][/tex]
This is equivalent to:
[tex]\[
y = -x + 7
\][/tex]
The equation in point-slope form that fits our requirements is:
[tex]\[
y - 5 = -(x - 2)
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{y - 5 = -(x - 2)}
\][/tex]
So, the correct solution is option 3.
