Perpendicular Lines and Slopes

Perpendicular lines are lines that intersect at right angles.

If you multiply the slopes of two perpendicular lines in the plane, you get 1 . That is, the slopes of perpendicular lines are opposite reciprocals.

(Exception: Horizontal and vertical lines are perpendicular, though you can't multiply their slopes, since the slope of a vertical line is undefined.)

We can write the equation of a line perpendicular to a given line if we know a point on the line and the equation of the given line.

Example :

Write the equation of a line that passes through the point ( 1,3 ) and is perpendicular to the line y=3x+2 .

Perpendicular lines are lines that intersect at right angles.

The slope of the line with equation y=3x+2 is 3 . If you multiply the slopes of two perpendicular lines, you get 1 .

3( 1 3 )=1

So, the line perpendicular to y=3x+2 has the slope 1 3 .

Now use the point-slope form to find the equation.

y y 1 =m( x x 1 )

We have to find the equation of the line which has the slope 1 3 and passes through the point ( 1,3 ) . So, replace m with 1 3 , x 1 with 1 , and y 1 with 3 .

y3= 1 3 ( x1 )

Use the distributive property.

y3= 1 3 x+ 1 3

Add 3 to each side.

y3+3= 1 3 x+ 1 3 +3 y= 1 3 x+ 10 3

Therefore, the line y= 1 3 x+ 10 3 is perpendicular to the line y=3x+2 and passes through the point ( 1,3 ) .