In the rectangular coordinate system, are the points (a, b) and (c, d) equidistant from the origin?
(1) a/b = c/d
(2) √(a²) + √(b²) = √(c²) + √(d²)
Statement 1: a/b = c/d
If the equation is 1/2 = 1/2, then (1,2) and (1,2) are equidistant from the origin.
If the equation is 1/2 = 2/4, then (1,2) and (2,4) are not equidistant from the origin.
INSUFFICIENT.
Statement 2: √(a²) + √(b²) = √(c²) + √(d²)
√x² = |x|.
Rephrasing the statement, we get:
|a| + |b| = |c| + |d|.
If the equation is |0| + |2| = |0| + |2|, then (0,2) and (0,2) are equidistant from the origin.
If the equation is |0| + |2| = |1| + |1|, then (0,2) and (1,1) are not equidistant from the origin.
INSUFFICIENT.
Statements 1 and 2 combined:
Let a/b = c/d = 2.
Then a=2b and c=2d.
Substituting a=2b and c=2d into |a|+|b| = |c|+|d|, we get:
|2b|+|b| = |2d|+|d|
3|b| = 3|d|
|b| = |d|, implying that |a| = |c|.
Since the x values in (a,b) and (c,d) are equidistant from the origin, and the y values in (a,b) and (c,d) are equidistant from the origin, the two points are equidistant from the origin.
By extension, if a/b = c/d = k, where k is any nonzero value, the result will be the same.
SUFFICIENT.
The correct answer is
C.
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