To determine the relationship between segments [tex]\(A B\)[/tex] and [tex]\(C D\)[/tex], we need to analyze the slopes of the lines on which these segments lie.
1. Equation of segment [tex]\(A B\)[/tex]:
The given equation is [tex]\(y - 4 = -5(x - 1)\)[/tex].
To find the slope, we rewrite this equation in the slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[
y - 4 = -5(x - 1)
\][/tex]
Distribute [tex]\( -5 \)[/tex] on the right-hand side:
[tex]\[
y - 4 = -5x + 5
\][/tex]
Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = -5x + 9
\][/tex]
From this equation, the slope [tex]\( m \)[/tex] of segment [tex]\( A B \)[/tex] is [tex]\( -5 \)[/tex].
2. Equation of segment [tex]\(C D\)[/tex]:
The given equation is [tex]\(y - 4 = \frac{1}{5}(x - 5)\)[/tex].
To find the slope, we rewrite this equation in the slope-intercept form [tex]\(y = mx + b\)[/tex].
[tex]\[
y - 4 = \frac{1}{5}(x - 5)
\][/tex]
Distribute [tex]\( \frac{1}{5} \)[/tex] on the right-hand side:
[tex]\[
y - 4 = \frac{1}{5}x - 1
\][/tex]
Add 4 to both sides to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{1}{5}x + 3
\][/tex]
From this equation, the slope [tex]\( m \)[/tex] of segment [tex]\( C D \)[/tex] is [tex]\( \frac{1}{5} \)[/tex].
3. Analyzing the slopes:
- The slope of segment [tex]\( A B \)[/tex] is [tex]\( -5 \)[/tex].
- The slope of segment [tex]\( C D \)[/tex] is [tex]\( \frac{1}{5} \)[/tex].
Two lines are perpendicular if the product of their slopes is [tex]\( -1 \)[/tex]. That is, if the slopes [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] satisfy [tex]\( m_1 \cdot m_2 = -1 \)[/tex].
Let's check the product of the slopes:
[tex]\[
-5 \cdot \frac{1}{5} = -1
\][/tex]
Since the product of their slopes is [tex]\( -1 \)[/tex], the lines are perpendicular.
Based on this, the correct statement that proves the relationship of segments [tex]\( A B \)[/tex] and [tex]\( C D \)[/tex] is:
They are perpendicular because they have slopes that are opposite reciprocals of -5 and [tex]\( \frac{1}{5} \)[/tex].
