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  3. P, Q, and R are three points in a plane, and R does not lie on line PQ. Which of

P, Q, and R are three points in a plane, and R does not lie on line PQ. Which of

游客2025-02-24  53
问题 P, Q, and R are three points in a plane, and R does not lie on line PQ. Which of the following is true about the set of all points in the plane that are the same distance from all three points?
选项 A、It contains no points.
B、It contains one point.
C、It contains two points.
D、It is a line.
E、It is a circle.
答案 B
解析 First consider just two of the three points, say P and Q, and the set of points in the plane that are the same distance from them. Clearly the midpoint of line segment PQ is such a point. Are there others? You may recall from geometry that the points on the line that bisects PQ and is perpendicular to PQ are all the points that are equidistant from P and Q. Similarly, the points in the plane that lie on the perpendicular bisector of line segment PR are all the points that are equidistant from points P and R.
    Because R does not lie on line PQ, line segments PQ and PR do not lie on the same line, and so their respective perpendicular bisectors are not parallel. Therefore, you can conclude that the two perpendicular bisectors intersect at a point. The point of intersection is on both perpendicular bisectors, so it is equidistant from P and Q as well as from P and R. Therefore, the point of intersection is equidistant from all three points. Are there any other points that are equidistant from P, Q, and R ? If there were, they would be on both perpendicular bisectors, but in fact only one point lies on both lines. The correct answer is Choice B.
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