Writing equations of perpendicular lines (example 2) | High School Math | Khan Academy

Find the equation of a line perpendicular to this line that passes through the point (2, 8).

So this first piece of information, that it's perpendicular to that line right over there, what does that tell us? Well, if it's perpendicular to this line, its slope has to be the negative inverse of two-fifths. So its slope, the negative inverse of two-fifths, the inverse of two-fifths is five.

Let me do it in a better color, a nicer green. If this line's slope is negative two-fifths, the equation of the line we have to figure out that's perpendicular, its slope is going to be the inverse. So instead of two-fifths, it's going to be five-halves. Instead of being a negative, it's going to be a positive.

So this is the negative inverse of negative two-fifths, right? You take the negative sign, it becomes positive. You swap the five and the two, you get five-halves. So that is going to have to be our slope.

And we can actually use the point-slope form right here. It goes through this point right there, so let's use point-slope form:

y minus this y value, which has to be on the line, is equal to our slope, five-halves, times x minus this x value, the x value when y is equal to 8.

And this is the equation of the line in point-slope form. If you want to put it in slope-intercept form, you can just do a little bit of algebra, algebraic manipulation.

y minus 8 is equal to, let's distribute the five-halves. So five-halves x minus five-halves times 2 is just 5.

Then add 8 to both sides, you get y is equal to five-halves x, add 8 to negative 5, so plus 3. And we are done.