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In the xy‐coordinate plane, is point R equidistant from points (–3,–3) and (1,–3) ?

1. The x‐coordinate of point R is –1.

2. Point R lies on the line y = –3.


    A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

    B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

    C. BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

    D. EACH statement ALONE is sufficient.

    E. Statements (1) and (2) TOGETHER are NOT sufficient.

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答案:
A
="font-size: 14px;">Algebra Coordinate Geometry

1. Since the x-coordinate of R is –1, R lies on the line with equation x = –1. The figure below shows points A (–3,–3) and B (1,–3), the line x = –1, and R with x-coordinate –1 and arbitrary y-coordinate y.

image.png


Since the point (–1,–3) is the midpoint of the horizontal segment image.png and the vertical line x = –1 contains (–1,–3), the line x = –1 is the perpendicular bisector of image.png. Every point on the perpendicular of image.pngis equidistant from A and B. Since R is on the line x =–1, R is equidistant from A and B.

Alternatively, by the distance formula, the distance between (–3,–3) and R is

image.png


Also by the distance formula, the distance between (1,–3) and R is

image.png

Since the distance between (–3,–3) and R is the same as the distance between (1,–3) and R, it follows that R is equidistant from (–3,–3) and (1,–3); SUFFICIENT.

2. Given that R is on the line y = –3, the y-coordinate of R is –3. Since the x-coordinate of R is not specified, R could be at multiple locations on the line y = –3, as shown in the chart and figure below; NOT sufficient.


x-coordinate

distance from (–3,–3)

distance from (1,–3)

equidistant?

–1

2

2

yes

3

6

2

no

image.png


The correct answer is A; 
statement 1 alone is sufficient.

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