I have seen several questions that seem to test the same principle, but which I always miss. They all are something like this: What is the distance from some point (x,y) to some line on the coordinate plane. I have also seen this version: what is the distance from a circle with the center at the origin and radius of 1 to some line. I know the distance formula but do not know how to calculate distance from a point to a line, or how to figure out where the shortest distance may be.
I'd point out that while I've seen the occasional prep company question that asks about distances from points to lines, I've never, in ten years in this field, seen a real GMAT question that does. So while memorizing formulas like the one in the posts above might help you on a high school coordinate geometry test, it is extremely unlikely to make any difference on the GMAT.
And if you were given some point (a, b) and some line y = mx + c, and asked to find the perpendicular distance from (a,b) to the line, you could always:
* find the equation of the line perpendicular to y = mx + c which contains (a,b). That line has a slope of -1/m, and you can find its y-intercept by plugging in the point
* now find where the first line and its perpendicular intersect by solving the two equations together - that will be at some point (d,e)
* find the distance from (d,e) to (a,b) to get the answer
Following those steps, you can derive the formula in the posts above. You may well need to do one of the three things above in a question, but you're very unlikely to need to do all three, so I think it's vastly preferable to learn each concept individually rather than memorizing formulas that you'll most likely never have occasion to use.
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