Maintaining Mathematical Proficiency
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Parallel lines have the same slope. However, the product of the slopes of two perpendicular lines is -1.
Parallel Lines: a∥ b, c∥ d
Perpendicular Lines: a⊥ c, a⊥ d, b⊥ c, b⊥ d
Two lines are parallel if their slopes are identical. To tell if two lines are perpendicular, we check if a product of their slopes is -1. Let's tackle each of these questions one at a time.
For this exercise we have been given two points on each line, so we have enough information to calculate their slopes using the Slope Formula.
m=y_2- y_1/x_2- x_1
Note that when choosing points to substitute for ( x_1, y_1) and ( x_2, y_2), it does not matter which points on the line we choose since the result will be the same. Let's start with line a which passes through ( 0, -3) and ( 3, 1).
Substitute ( 0,-3) & ( 3,1)
a-(- b)=a+b
Add and subtract terms
The slope of line a is 43. We can use the same method to identify the slopes of lines b, c, and d.
| Line | Point 1 | Point 2 | y_2-y_1/x_2-x_1 | Slope |
|---|---|---|---|---|
| a | ( 0, -3) | ( 3, 1) | 1-( -3)/3- 0 | 4/3 |
| b | ( -3, -3) | ( 0, 1) | 1-( -3)/0-( -3) | 4/3 |
| c | ( -2, 4) | ( 2, 1) | 1- 4/2-( -2) | -3/4 |
| d | ( -4, 2) | ( 4, -4) | -4- 2/4-( -4) | -3/4 |
Now that we have identified the slope of each line, we can see that a and b as well as c and d have the same slope which means they are parallel. For lines with different slopes, we can conclude that they are not parallel.
To determine whether or not the lines are perpendicular, we calculate the product of their slopes. Any two lines with the slopes whose product equals -1 are perpendicular. m_1* m_2 ? =-1 Let's substitute the slopes that we found, 43 and - 34.
m_1= 4/3, m_2= -3/4
a(- b)=- a * b
a/a=1
This means that all the lines with slope 43 are perpendicular to the lines with slope - 34. Therefore, in our case a and b are both perpendicular to c and d.
Let's summarize what we found. Parallel Lines:& a∥ b, c∥ d Perpendicular Lines:& a⊥ c, a⊥ d, & b⊥ c, b⊥ d