Question

find the distance from the point (2,-1) to the line y=4

Answer 1

Well when looking at a graph like the one attached, the y axis is the only distance worth focusing on because the line runs parallel with the x axis. the line is 4 above the x axis and the point is 1 below so subtract the distances to find the difference. 4--1 = 5

Answer 2

Great solution, @lasttccasey, especially since you've provided an excellent illustration.
However, I'd suggest the following wording for your written statement:
"As shown in the attached graph, we're only interested in the measure of a distance parallel to the y-axis. ... 4-(-1) = 5."

Answer 3

don't forget an important point, the shortest distance will always be on the line perpendicular to the line you are talking about.

Perpendicular lines have a slope that is the negative reciprocal of the slope of the other line.
if the first line is:
\[ y = \frac{2}{3}x + 3 \]
then the second will be:
\[ y = -\frac{3}{2}x + 3\]

Really you wan to find a the perpendicular line that the point goes through to solve this problem generally.

Answer 4

in this case x=2 is the line perpendicular, so its pretty simple.

Answer 5

Thank you guys soo much!

Answer 6

to find distance you need to first find a perpendicular line to y=4 that passes through the point (2,-1)

Answer 7

a line perpendicular to y=4 is in the form
x=k
so for 2,-1 to be a solution to x=k
then 2=k, there is no place to substitute the -1 in you can think of this as
x=k + 0y <--- even if you plug in -1 for y it wont change the value because of that 0

Answer 8

While it's true that the distance in question lies on the line which is perpendicular to y=4 and which also passed through (2,-1), I submit that it is unnecessary to find the equation of this perpendicular line. @lasttccasey 's excellent drawing provides all the info needed. As @polaris_s0i has pointed out, the shortest distance from any point to a given line lies along a perpendicular to the given line. By inspection of the drawing, that line is x=-2. Simply count the number of scale divisions between the point (2,-1) and the line y=4; that's the distance between the given point and the given line.

Answer 9

Please excuse my typos: "passed" should be "passes," and "that line is x=-2" should be "that line is x=2." Oops.

Answer 10

There is definitely nothing wrong with using you're intuition to bypass unneeded math for sure @mathmale :). In fact that is the best thing to do!

I figured that it might be good to know how to do this in general.