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 L₁ and L₂ in figure 1-5. The slope and angle of inclination of L₁ are m₁ and a₁, respectively. The slope and angle of inclination of L₂ are m₂ and a₂, respectively. Then the following is true:

Although not shown here, the fact that a₂ (fig1-5) is equal to a, plus 90 can be proven geometrically. Because of this relationship

Figure 1-5.-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 1-6 show that line L₁ is perpendicular to line L₂. Line L₁ passes through points P₁(0,5) and P₂ (^-1,3). Line L₂ passes through points P₂(-1,3) and P₃(3,1).

Figure 1-6.-Proving lines perpendicular.

SOL UTION: Let m₁ and M₂ represent the slope of lines L₁ and L₂, respectively. Then we have

Since their slopes are negative reciprocals of each other, the lines are perpendicular.

PRACTICE PROBLEMS:

1. Find the distance between P₁ (5,3) and P₂ (6,7).

2. Find the distance between P₁ (1/2,1) and P₂ (3/2,5/3).

3. Find the midpoint of the line connecting P₁ (5,2) and P₂ (-1, -3).

4. Find the slope of the line joining P₁ (-2,- 5) and P₂ (2,5).

5. Find the slope of the line perpendicular to the line joining P₁(-3,6) and P₂(-5,-2).

ANSWERS: 1, 17