This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Two Proofs Involving Isometries Theorem: The inverse of an isometry is an isometry.
Proof: Suppose that a : P + P is an isometry.
Let B = (1‘1 . Then, for any point X in P, [3(X) =X ifandonlyif a(X') = X, and, inthiscase, a°B(X) = X, since a°B(X) = a(B(X) ) = a(X ) = X.
Let X and Y be any two points in the plane P. [Weneedtoshowthat ( [3(X) B(Y)) = (XY).] Let x'= B(X) andlet Y'= B(Y). Then, a(X') =X and a(Y') =Y. V Sincea isanisometry, (a(X')a(Y')) = X'Y . V ' Thus,XY = X'Y', so, X Y = XY.
But, X'= B(X) andlet Y'= B(Y). Thus, (50‘) [3(Y)) = (XY) QED Problem: It is given that: __ ._ 6—)
AB is the diameter of circle C( Z , ZB ) . M is the midpoint of ZB and line s = AB . Line t is perpendicular to line 5 at point M . Line t intersects C(Z , ZB)at points H and K. Line t ToProve: (1)Rt(Z)=B and Rt(B)=Z. (2)RS(H)=K and RS(K)=H. Lines (3) 115(3) = B and Rs° RS(H) = H. Proof: (1) Since line t is the perpendicular bisector of segment ZB , R t ( Z) = B and Rt ( B ) = Z , by deﬁnition of a reﬂection about the mirror line t . 9—) (—) .— ..._....
(2) Since line s = AB and line t = HK , diamter AB is perpendicular to chord HK. By Theorem 4.5.4 ("If a diameter is perpendicular to a chord, then the diameter bisects it"), diameter AB bisects chord Thus, line 5 is the perpendicular bisector of R . Therefore, R s (H ) = K and R s ( K) = H , by deﬁnition of a reﬂection about the mrror line s .
(3) Since B is a point on line s , R s ( B ) = B , by deﬁnition of a reﬂection about the mrror line 5 .
By deﬁnition of composition of functions, R 5 ° R s (H ) = R s (R s ( H ) ) .
Asshownin(2), RS(H) = K. Therefore, Rs° RS(H) = RS(RS(H)) = RS(K) = H. Therefore, RS° Rs(H) = H. QED ...
View
Full
Document
This note was uploaded on 11/02/2010 for the course MATH modern geo taught by Professor Shirley during the Spring '10 term at University of Texas at Austin.
 Spring '10
 Shirley

Click to edit the document details