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What is the easiest way to find the line perpendicular to a given line?

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brandyboy | Student | eNotes Newbie

Posted October 6, 2010 at 5:16 PM via web

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What is the easiest way to find the line perpendicular to a given line?

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hustoncmk | High School Teacher | (Level 1) Assistant Educator

Posted October 6, 2010 at 7:43 PM (Answer #1)

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Lines that are perpendicular have slopes that are negative reciprocals of each other.  What this means is if the slope for the first line is a/b the perpendicular line would have a slope of -b/a.

Let's start with an example with numbers so that you can see the full process.

Find the line perpendicular to y = 1/3x + 2, that crosses at point (3,2)

It is easiest to start with a line in slope-intercept form. For our example we will use y = 1/3x + 2 as the original line. 1/3 is the slope so we know that the line goes up 1 unit and right 3 units. The 2 is the y-intercept, this tells us that this line crosses the y axis at (0,2).

To find the perpendicular line, the first step is to find the negative reciprical of the slope 1/3.  Invert the fraction to 3/1 and add a negative sign.  The new slope is -3. One point is given as (3,2). If you substitute these into the equation with the new slope you can solve for the slope intercept and complete the new equation.

y = -3x + b

2 = -3(3) + b

2 = -9 +b

11 = b

y = -3x + 11 (substitute the new slope and y-intercept into the equation) is perpendicular to y = 1/3x + 2.

I hope that this example helps you understand how to solve perpendicular lines in the future. 

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neela | High School Teacher | (Level 3) Valedictorian

Posted October 6, 2010 at 8:17 PM (Answer #2)

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If ax+by+c  is the equation of a line, to find the equation of tlene perpendicular to ax+by+c = 0 we just interchange the coefficients of x and y and put a minus sign to one of the two.

So the equation of the line perpendicular line to the line ax+by+c is of the form, bx-ay + k1 = 0 Or -bx+ay + k2 = 0.

This is because ax+by +c = 0 has a slope a/b.

So any any line  perpendicular to ax+by+c = 0, must have slope , m  which , with a/b gives a product -1. Or m*(a.b) = -1. So m = -b/a.

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