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SLOPES OF PARALLEL AND PERPENDICULAR LINES If two lines are parallel, their slopes must be equal.
Each line will cut the X axis at the same angle a so that if _{} We conclude that two lines which are parallel have the
same slope. Suppose that two lines are perpendicular to each other, as
lines L_{1} and
L_{2} in figure 15. The slope and angle of inclination
of L_{1} are m_{1 }and a_{1}, respectively.
The slope and angle of inclination of L_{2} are m_{2} and a_{2}, respectively.
Then the following is true:
Although not shown here, the fact that a_{2} (fig15)
is equal to a, plus 90 ° can be proven geometrically. Because of this
relationship
Figure
15.Slopes of perpendicular lines.
Replacing
tan a, and tan a_{Z} by their equivalents in terms of slope, we have
We
conclude that if two lines are perpendicular,
the slope of one is the negative reciprocal of the slope of the other.
Conversely, if the slopes of two lines are negative reciprocals of each other,
the lines are perpendicular.
EXAMPLE: In figure 16 show that line L_{1} is perpendicular to line L_{2}._{
}Line L_{1} passes through points P_{1}(0,5)
and P_{2} (^{}1,3). Line L_{2} passes through points P_{2}(1,3)
and P_{3}(3,1).
Figure 16.Proving lines perpendicular. SOL
UTION: Let m_{1} and M_{2} represent the slope of lines L_{1} and L_{2},_{ }respectively.
Then we have
Since
their slopes are negative reciprocals of each other, the lines are
perpendicular. PRACTICE
PROBLEMS: 1.
Find the distance between P_{1} (5,3) and P_{2} (6,7). 2.
Find the distance between P_{1} (1/2,1) and P_{2} (3/2,5/3). 3. Find the midpoint of the line connecting P_{1} (5,2) and P_{2}
(1, 3). 4.
Find the slope of the line joining P_{1} (2, 5) and P_{2} (2,5). 5. Find the slope of the line perpendicular to the line joining P_{1}(3,6)
and P_{2}(5,2). ANSWERS:
1, 17
