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Suppose ABCD is a rectangle, O is the center of the circle. If the coordinates o
Suppose ABCD is a rectangle, O is the center of the circle. If the coordinates o
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2024-01-12
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问题
Suppose ABCD is a rectangle, O is the center of the circle. If the coordinates of O are (1, 2), coordinates of A are (0, 3), and coordinates of D are (4, 7) then what are the coordinates of C? Suppose ABCD is a rectangle, O is the center of the circle. If the coordinates of O are (-1, 0), coordinates of A are (-2, 1), and coordinates of D are (3, 6) then what are the coordinates of C?
选项
A、(5, 6)
B、(6, 5)
C、(6, 4)
D、(4, 6)
E、(5, 4)
答案
E
解析
这个题目按照《GRE考试官方指南》的分类来说是一道解析几何的题目。如果按照解析几何的方法来解决这个问题的话,我们可以采取以下步骤,已知A点坐标、O点坐标,求得直线AO方程。又因为B在直线AO上,利用圆半径相等还知道|AO|=|OB|,解得B点坐标,同时,已知A点坐标、D点坐标,求得直线AD方程,因为直线AD平行于BC,所以可求直线BC斜率,已求得B点坐标,所以直线BC方程可求,再利用矩形对边相等。|AD|=|BC|,求得C点坐标。
向量是一个带有方向的线段。向量相等要求两个条件:一是方向相同,共线或平行,二是距离相等。如果两个向量相等,会有一个非常好的性质,就是两个向量端点对应坐标的差相等。
如上图所示,我们知道,
(AO,0B都为半径,且A、O、B三点共线)。于是A(-2,1)→D(-1,0),横坐标加1,纵坐标减1,于是从O到B,同样横坐标加1,纵坐标减1,0(-1,0)→B(0,-1)。这样就很轻松地求出了B点坐标。
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