For #18, can someone please explain if the lines are parallel, perpendicular, or neither?  Thanks!

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lemjay's profile pic

lemjay | High School Teacher | (Level 3) Senior Educator

Posted on

To determine if the two lines are parallel, perpendicular or neither, solve the slope of each line. To do so, use the formula:

`m = (y_2-y_1)/(x_2-x_1)`

When their slopes are already known, consider these following conditions.

> If the slopes of the two lines are the same, then they are parallel.

    `m_1=m_2`

> If the product of the slopes of the two lines is equal to -1, then they are perpendicular.

    `m_1*m_2=-1`

#18
Line 1: through (3,-1) and (6,-4)
Line 2: through (-4,5) and (-2,7)

Slope of Line 1 is:

`m_1= (-4-(-1))/(6-3)=(-3)/3`

`m_1=-1`

Slope of Line 2 is:

`m_2=(7-5)/(-2-(-4))=(2)/(2)`

`m_2=1`

Since their slopes are not the same, multiply the two slopes to determine if it satisfy the condition for perpendicular lines.

`m_1*m_2=-1*1=-1`

Hence, the two lines are perpendicular.

#20
Line 1: through (-1,4) and (2,5)
Line 2: through(-6, 2) and (0,4)

Slope of Line 1 is:

`m_1= (5-4)/(2-(-1))`

`m_1=1/3`

Slope of Line 2 is :

`m_2=1/3`

Therefore, the two lines parallel.

#22
Line 1: through (-3,2) and (5,0)
Line 2: through ( -1, -4) and (3,-3)

Slope of Line 1:

`m_1=(0-2)/(5-(-3))=(-2)/8`

`m_1=-1/4`

Slope of line 2:

`m_2=(-3-(-4))/(3-(-1))`

`m_2=1/4`

Since their slopes are not the same, multiply the two slopes to determine if it satisfy the condition for perpendicular lines.

`m_1*m_2=-1/4*1/4=-1/16`

Thus, the two lines are neither parallel nor perpendicular.

cjpierce's profile pic

cjpierce | Middle School Teacher | (Level 1) Adjunct Educator

Posted on

Absolutely! 

Identifying a parallel or perpendicular set of lines is all about the slope of the line. Parallel lines have exactly the same slope, meaning they will never intersect with one another. Perpendicular lines intersect at a 90 degree angle, which means the slopes are opposite reciprocals. For example, if the slope of one line is 2, the slope of the other is -(1/2). The signs are opposites and the numbers are flipped.

So, to answer the question we need to figure out the slope of the two lines. Slope is the change in y over the change in x. Let's look at Line 1 first...

The coordinates are (3, -1) and (6, -4).

The y values are the second numbers, in this case -1 and -4. What happened to get from -1 to -4? The number went down by 3, so this gives us a -3 on top of the slope (it's the change in y)...

The x values are the first numbers, in this case 3 and 6. These values went up by 3, so the bottom of our slope is +3. 

Therefore, our slope is -3/3, which is -1.

Now let's look at Line 2...

The y values go up by 2 (from 5 to 7) and the x values go up by 2 (from -4 to -2). Therefore, the slope of Line 2 is 2/2, which is 1.

Since the slopes are -1 and 1, they are definitely not parallel lines because they don't have the same slope. However, they ARE perpendicular lines, because the recipricol of 1 is still 1, and the signs are opposites (one is positive, the other is negative).

Therefore....these lines are perpendicular!

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