ERIC Number: EJ875464
Record Type: Journal
Publication Date: 2004
Abstractor: As Provided
Reference Count: 0
The Centers of Similarity of Two Non-Congruent Squares
Ayoub, Ayoub B.
AMATYC Review, v25 n2 p23-28 Spr 2004
The topic of centers of similarity can be treated synthetically or analytically. While the synthetic method is more practiced, the analytic approach is more appropriate when the problem is given in an analytic geometry setting. In this article, two non-congruent squares ABCD and A'B'C'D' are given, where A(0,0), B(3,0), C(3,3), D(0,3) and A'(5,4), B'(9,7) C'(6,11) D'(2,8). When AB [grouped] is mapped onto each of A'B' [grouped], B'C' [grouped], C'D' [grouped], and D'A' [grouped], four direct similarities are determined. These are dilative rotations which map ABCD and A'B'C'D'. On the other hand, four opposite similarities are determined if AB [grouped] is mapped onto each of A'D' [grouped], D'C' [grouped], C'B' [grouped], and B'A' [grouped]. These are dilative reflections which map ABCD onto A'B'C'D'. To determine the eight similarities, the following theorem is used: Every similarity transformation with ratio k has the equations: x' = ax + by + c and y' = plus or minus (-bx + ay) + d, where a[superscript 2] + b[superscript 2] = k[superscript 2], the plus sign corresponds to a direct similarity and the minus sign corresponds to an opposite similarity. The center (x,y) of such similarity is obtained by setting (x',y'). For the above eight similarities, the centers of similarity are: (-51/10, -33/10), (3/2, 9/2), (225/58, 177/58), (6, -3/2), (45/8, -33/8), (-57/8, 3/8), and (-3, -33/4). There are only seven centers because the center (3/2, 9/2) belongs to two similarities, one direct and the other opposite. It has been shown that the seven centers of similarity lie on the circle 8x[superscript 2] + 8y[superscript 2] + 12x + 30y - 333 = 0.
Descriptors: Transformations (Mathematics), Educational Technology, Geometry, Geometric Concepts, Mathematics Instruction, College Mathematics, Community Colleges
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Publication Type: Journal Articles; Reports - Descriptive
Education Level: Two Year Colleges
Authoring Institution: N/A