Chapter Test
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Parallel lines have the same slope. The slopes of perpendicular lines are negative reciprocals.
Parallel Lines: None of the lines are parallel.
Perpendicuar Lines: Lines 1 and 2 are perpendicular.
Two lines are parallel if their slopes are identical. To tell if two lines are perpendicular, we have to check if their slopes are negative reciprocals. Let's tackle these questions one at a time.
For this exercise, we have been given equations that are not in slope-intercept form, so let's first rewrite all of them to identify their slopes.
| Line | Given Equation | Slope-intercept form | Slope |
|---|---|---|---|
| 1 | y-c=ax | y=ax+c | a |
| 2 | ay=- x-b | y=-1/ax-b/a | -1/a |
| 3 | ax+y=d | y=- ax+d | - a |
For lines with different slopes, we can conclude that they are not parallel. To determine whether or not they are perpendicular, we will calculate the product of their slopes. Any two slopes whose product equals -1 are negative reciprocals, and therefore perpendicular. Let's start by checking lines 1 and 2.
m_1= a, m_2= -1/a
Multiply
a/a=1
Therefore, lines 1 and 2 are perpendicular. We will use a similar method to check if lines 1 and 3 or 2 and 3 are perpendicular.
| Lines | Slope 1 | Slope 2 | Product |
|---|---|---|---|
| 1 and 2 | a | -1/a | -1 |
| 1 and 3 | a | - a | - a^2 |
| 2 and 3 | -1/a | - a | 1 |
We have found that only lines 1 and 2 are perpendicular to one another.