Question

#3 please

Isometries 1. (Apt) Prove that the composition of two isometries is an isometry. 2.4pt) Prove that if F and G is an isometry of the plane and F(AABC) = G(AABC) = AA'B'C', then F = G. That is, an isometry is uniquely determined by three non-collinear points (a triangle) and their images. Consider using an inverse of one of the isometries. 3. (4pt) To prove that a reflection, Rom, is an isometry, it must be shown that it preserves distance between any pair of points in the plane. Let P and Q be two points in the plane such that PQ intersects m at a point R between P and Q. Prove that the distance PQ is preserved under R.

Answer 1

solution- Port of Let Ti and To be isometries. Ti (A) = À and Tp (B] = B. Therefore by Drometry Tue AB = A B And, T2 (A) = A and T2 (B) = 8 Therefore by bsometry Tag AB = A B Tzott - Tz( 11 (4)] = Tel A') = A" Toot To (To(A)] = To (B') = 8" Therefore under composition AB = A" B - distance is preferred Hence, the composition of two isometries is an isometry.