To determine which equation represents a line that is perpendicular to line [tex]\( KL \)[/tex] with the equation [tex]\( y = 4x + 5 \)[/tex], we must first understand the concept of perpendicular slopes. If two lines are perpendicular, the slopes of these lines are negative reciprocals of each other.
1. Identify the slope of line [tex]\( KL \)[/tex]:
The given equation of line [tex]\( KL \)[/tex] is [tex]\( y = 4x + 5 \)[/tex]. This is in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope. For line [tex]\( KL \)[/tex], the slope [tex]\( m \)[/tex] is [tex]\( 4 \)[/tex].
2. Calculate the perpendicular slope:
The slope of any line perpendicular to another is the negative reciprocal of the slope of the original line. The slope of line [tex]\( KL \)[/tex] is [tex]\( 4 \)[/tex], so the negative reciprocal is:
[tex]\[
-\frac{1}{4}
\][/tex]
3. Examine the given options for the perpendicular slope:
Let’s go through each of the options to find the equation with the perpendicular slope of [tex]\( -\frac{1}{4} \)[/tex]:
- [tex]\( y = 4x - 8 \)[/tex]: The slope here is [tex]\( 4 \)[/tex] (same as the slope of [tex]\( KL \)[/tex], not perpendicular).
- [tex]\( y = \frac{1}{4}x - 8 \)[/tex]: The slope here is [tex]\( \frac{1}{4} \)[/tex] (not the negative reciprocal of [tex]\( 4 \)[/tex]).
- [tex]\( y = -4x - 8 \)[/tex]: The slope here is [tex]\( -4 \)[/tex] (not the negative reciprocal of [tex]\( 4 \)[/tex]).
- [tex]\( y = -\frac{1}{4}x - 8 \)[/tex]: The slope here is [tex]\( -\frac{1}{4} \)[/tex], which is the negative reciprocal of [tex]\( 4 \)[/tex].
4. Conclusion:
The equation for a line that is perpendicular to line [tex]\( KL \)[/tex] is:
[tex]\[
y = -\frac{1}{4}x - 8
\][/tex]
Thus, the correct option is:
[tex]\[ y = -\frac{1}{4}x - 8 \][/tex]
