To determine the relationship between segments [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex], we need to analyze the slopes of the lines on which these segments lie.
First, let's find the slope of the line on which segment [tex]\( AB \)[/tex] lies. The equation for this line is given by:
[tex]\[ y - 4 = -5(x - 1) \][/tex]
We can rewrite this equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 4 = -5(x - 1) \][/tex]
[tex]\[ y - 4 = -5x + 5 \][/tex]
[tex]\[ y = -5x + 9 \][/tex]
Thus, the slope [tex]\( m \)[/tex] of the line is:
[tex]\[ m_{AB} = -5 \][/tex]
Now, let's find the slope of the line on which segment [tex]\( CD \)[/tex] lies. The equation for this line is given by:
[tex]\[ y - 4 = \frac{1}{5}(x - 5) \][/tex]
Again, we'll rewrite this equation in slope-intercept form [tex]\( y = mx + b \)[/tex]:
[tex]\[ y - 4 = \frac{1}{5}(x - 5) \][/tex]
[tex]\[ y - 4 = \frac{1}{5}x - 1 \][/tex]
[tex]\[ y = \frac{1}{5}x + 3 \][/tex]
Thus, the slope [tex]\( m \)[/tex] of the line is:
[tex]\[ m_{CD} = \frac{1}{5} \][/tex]
To determine if the lines are perpendicular, we check if their slopes are opposite reciprocals. For slopes to be opposite reciprocals, the product of the slopes must be [tex]\(-1\)[/tex]:
[tex]\[ m_{AB} \times m_{CD} = -5 \times \frac{1}{5} = -1 \][/tex]
Since the product of the slopes is [tex]\(-1\)[/tex], we conclude that the lines are indeed perpendicular. Therefore, the segments [tex]\( AB \)[/tex] and [tex]\( CD \)[/tex] are perpendicular.
The correct statement is:
They are perpendicular because they have slopes that are opposite reciprocals of -5 and [tex]\(\frac{1}{5}\)[/tex].
