Official Guide Explanation:
Data Sufficiency #164




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Solution and Metadata

Question: 164
Page: 288
Difficulty: 7 (Very Difficult)
Category 1: Geometry > Coordinate Geometry > Distance
Category 2: Word Problems > Geometry Problems >
Category 3: Arithmetic > Powers and Roots of Numbers > Powers

Explanation: Statement (1) is insufficient: it includes nothing about u or v. Statement (2) is also insufficient. It may be easier to work with these equations (at least for now) in the form u + r = 1 and v + s = 1. If u and s are 1 and r and v are 0, yes, the points are equidistant from the origin. There are plenty of counterexamples, though: for instance if u and v are 0 and r and s are 1.

Taken together, the statements are sufficient. Note that the equation in (1) can be rewritten in two ways:

r = 1 - s

s = 1 - r

The first indicates that, since v = 1 - s, r = v. The second indicates that, since u = 1 - r, s = u. Therefore, the two sets of coordinates are equal, though the x and y coordinates are reversed. If the distance of each individual coordinate is equal to a coordinate in the other point, the distance of the points from the origin is the same. Choice (C) is correct.

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