# Are the two lines L1 through (-1,4) and (0,4) and L2 through (5,6) and (5,2) parallel, perpendicular or neither?

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To determine if lines are parallel, perpendicular or neither, we need o determine each line's slope.

If slopes are equals then they are parallel.

Let us calculate L1 slope (m1)

m1= (y2-y1)/(x2-x1) = (4-4)/(0+1)= 0

slope 0 means that the line L1 is horizontal or parallel to the x-axis.

Now the slope for L2 (m2):

m2= (y2-y1)/(x2-x1)= (2-6)/(5-5)= -4/0 Undefined slope

Undefined slopes means that the line L2 is vertical or parallel with y-axis

Then the lines L1 and L2 are perpendicular.

The two lines will be parallel if their slope is same.

Slope of a line AB, passing through point A (x1, y1) and B (x2, y2) is given by:

Slope = m =(y2 - y1)/(x2 - x1)

Using this equation we calculate the slope of given lines L1 and L2 as follows.

Slope of L1 - m1 = (4 - 4)/(0 + 1) = 0/1 = 0

Slope of L2 - m2 = (2 - 6)/(5 - 5) = -4/0 = - Infinity

As the slopes of the lines L1 and L2 are not same, they are not parallel.

From the values of the slopes obtained, we can infer that line L1 is parallel to x-axis, while line L2 is parallel to y-axis. Thus the two lines are perpendicular to each other.

Based on fact that if 2 lines are parallel, their slopes are equal and if the 2 lines are perpendicular, the product of their slopes is -1, we'll calculate their slopes first and we'll check the relation between slopes.

To calculate the slope of the line L1, we'll use the formula:

m1 = (y2-y1)/(x2-x1)

Now, we'll substitute the coordinates:

m1 = (4-4)/(0+1)= 0/1 = 0

We'll calculate the second slope:

m2 = (2-6)/(5-5) = -4/0 = -inf.

That means that the slope is a line which is parallel to y axis.

Since the slope m1 represents a line that is parallel to x axis, that means that the slopes are perpendicular and the lines L1 and L2 are perpendicular, too.

The equations of a line in slope intercept form is:

y = mx+c since this pass through the points(-1,4) and (0,4) they must satisfy y =mx+c. Or

4 = m(-1)+c and

4 = m(0)+c

Subtrating, 4-4 = m(-1-0). Or m = 0/4 = 0 slope . So L1 is parallel to X axis

For the 2nd line the point (5,6) and (5,2) should satisfy y = nx+k. Or

6= 5n+k and

2=5n+k

Subtracting, 6-2= (5-5)n+0. Or n = 4/0 = infinity. So the line L2 is parallel to Y axis.

Therefore L1 ||X axis and L2 ||Y axis. So L1 and L2 are perpendicular to each other.