Mike@Magoosh wrote:

Mission2012 wrote:In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?

(1) a/b = c/d

(2) (a^2)^1/2 + (b^2)^1/2 = (c^2)^1/2 + (d^2)^1/2

I'm happy to help.

Statement #1 says, essentially, the lines have the same slope ----- if a/b = c/d, then b/a = d/c, and those are the slopes of lines through the original and each of these points. In other words, the two points lie on the same line through the origin. Interesting, but it doesn't answer the prompt question. This statement, alone and by itself, is insufficient.

Statement #2 seems funky --- this simplifies to
|a| + |b| = |c| + |d|
If all four numbers were positive, this would say (a + b) = (c + d), which means the points would lie on the same oblique line with a slope of -1, a line of the form y + x = k.
If the four numbers have different, then the point lie on such lines that are the reflection of each other in the various quadrants. Even if all the numbers are positive, and all points are in Q1, they may or may not be equidistant from the origin. This statement, alone and by itself, is insufficient.

Combined --- now, both are on the same line through the origin, and either they are in the same quadrant, in which case they are in the same place (a = c, b = d), or the two points are images of each other in 180 degree rotation around the origin, in which case (a = -c, b = -d), and in either one of these cases, the two points have to be equidistant from the origin. Combined, statements are sufficient.

Answer = [spoiler]C[/spoiler]

Let me know if you have any further questions.
Mike