Parallel and Perpendicular Lines for Algebra Dummies

Equations of parallel and perpendicular lines

Many freelancers practice a "trick" that tends to drive the Antisocial Network researchers bonkers. This trick is what's sometimes called the "If you can't dazzle 'em with your brilliance, baffle 'em with bullshit!" solution for writing about unfamiliar topics. Well, we found another example of that in today's DotD nominee, Grant D. McKenzie. The aeronautical engineering student chose to write about "How to Write Equations of Parallel & Perpendicular Lines" for eHow.com; the post has since been niched at Sciencing.com.

We checked with our algebra guru, and she told us that it's easy: "Parallel lines have the same slope but different X- and Y-intercepts. The slopes of perpendicular lines are negative reciprocals of each other." Grant, not surprisingly, managed to get the first half of that answer correct, although his route to a solution is rather hard to follow:

• "Write the equation for the first line and identify the slope and y-intercept.
• Example: y = 4x + 3 m = slope = 4 b = y-intercept = 3
• Copy the first half of the equation for the parallel line. A line is parallel to another if their slopes are identical.
• Example: Original line: y = 4x + 3 Parallel line: y = 4x
• Choose a y-intercept different from the original line."

We think the equations would be easier to decipher if Grant and his editor had added some punctuation. Grant followed by telling his readers that,

"Regardless of the magnitude of the new y-intercept, as long as the slope is identical, the two lines will be parallel."

While true, McKenzie's use of the word "magnitude" isn't quite correct: he really should have said, "value." That's just quibbling, though. It's when he gets to writing a perpendicular that Grant got in real trouble.

The real answer is that a perpendicular to a line of slope m has a slope of -1/m. It's that simple. Since his line's formula is y = 4x+3, the slope is 4. The negative reciprocal of 4 is -¼, so the equation must be y = -x/4 + 3, or 4y = -x + 12. These two lines will cross at the intercept; to get them to cross at any other point change the value of b from 12. McKenzie's instructions? Puzzle through this stuff:

Example: y = 4x + 3 m = slope = 4 b = y-intercept = 3 Transform for the "x" and "y" variable. The angle of rotation is 90 degrees because a perpendicular line intersects the original line at 90 degrees. Example: x' = x_cos(90) - y_sin(90) y' = x_sin(90) + y_cos(90) x' = -y y' = x Substitute "y'" and "x'" for "x" and "y" and then write the equation in standard form.

We're still trying to figure out the significance of the second and third steps, though we suspect he accidentally stuck part of the proof in there. We suspect, Grant couldn't figure it out, either. Whatever the case, McKenzie took an easy answer and made it hard. That's classic Dumbass of the Day work!

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