OA C

**Quote:**

To find the distance from the origin, we simply take the square root of the sum of the squared x- and y-coordinates, i.e. SQRT(x^+y^2).

(1) INSUFFICIENT: This simply tells us that the proportions between the x- and y-coordinates of both points are the same. E.g. take a = 5, b = 10, c = 6 and d = 12. The proportions are the same but the coordinate points are not the same distance from the origin. Conversely, if a = 5, b = 10, c = -5 and d = -10, then the proportions are equal and the coordinate points are the same distance from the origin.

(2) INSUFFICIENT: By simplifying the expression, we get |a| + |b| = |c| + |d|. This is not enough to tell if the points are equidistant. E.g. take a = 11, b = 1, c = 6 and d = 6. The expression |a| + |b| = |c| + |d| is true but the coordinate points are not the same distance from the origin. Conversely, if a = -6, b = 6, c = 6 and d = 6, then the given expression is true and the coordinate points are the same distance from the origin.

(1) AND (2) SUFFICIENT: Together the statements are sufficient. Why? If we know the proportion of a to b is the same as c to d and that |a| + |b| = |c| + |d|, then it must be the case that |a| = |c| and |b| = |d|. Plugging this into our distance formula, we get:

SQRT(a^+b^2) = SQRT(c^+d^2) ; Plug in |a| = |c| and |b| = |d| to get:

SQRT(a^+b^2) = SQRT(a^+b^2)

This is enough to show that the two points are equidistant.

The correct answer is C.