To answer this question, we first need to remember that:
• Two lines are parallel if and only if their slopes are equal, that is:
Two lines are perpendicular if and only if their slopes fulfil:
Hence, we need to find the slopes for each of the lines give. To do this we will write them in slope-intercept form:
where m is the slope and b is the y-intercept.
Line 1.
This line is already written in slope intercept form, comparing it with the equation above we conclude that the slope of line 1 is 3/4, that is:
Line 2.
Let's write line 2 in slope intercept form:
Comparing the last line with the slope-intercept equation we conclude that line 2 has slope 3/4, that is:
Line 3.
Let's write line 3 in the appropriate form:
From the last equivalent equation, we conclude that the slope of line 3 is equal to -4/3, that is:
Now, that we know each slope we can determine which lines are parallel, perpendicular or neither.
Line 1 and Line 2.
We notice that the slopes of lines 1 and 2 are equal since both of them are 3/4, this means that these lines are parallel.
Line 1 and Line 3.
Since the slope are different, we will check if they are perpendicular, let's use the condition stated above:
Since, the condition is fulfilled we conclude that lines 1 and 3 are perpendicular.
Line 2 and Line 3.
It is clear that these slopes are not equal, let's check for if they are perpendicular:
Hence, lines 2 and 3 are perpendicular.
Note: Since lines 1 and 2 are parallel and lines 1 and 3 are perpendicular this readily implies that lines 2 and 3 are perpendicular as well.
Summing up:
Line 1 and Line 2: Parallel
Line 1 and Line 3: Perpendicular
Line 2 and Line 3: Perpendicular