Affine Geometry and Euclidean Geometry

oevggurtvatre
timer Asked: Oct 5th, 2017

Question Description

All of these questions have to do with proving college level geometry. Please help ASAP.

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01:640:435 SECTION 01 GEOMETRY: HW3 (DUE DATE: OCT 5) (1) (SSS) Let ∆ = ∆ABC, ∆0 = ∆A0 B 0 C 0 be two triangles such that |AB| = |A0 B 0 |, |BC| = |B 0 C 0 | and |CA| = |C 0 A0 |. Prove that the two triangles ∆, ∆0 are Euclidean-congruent: that is, there is a rigid motion f ∈ Grm so that f (∆) = ∆0 . (Hint: first find a rigid motion f1 which sends AB to A0 B 0 . Then note that C, C 0 must lie in an intersection of two circles). (2) Consider the plane R2 and Rot the subgroup of Bij(R2 ) given by: f ∈ Rot if and only if f (~x) = R~x + p0 2 for some p0 ∈ R and R is a rotational matrix. That is,   cos θ − sin θ R= for some θ ∈ R. sin θ cos θ (a) Note that Rot ⊂ Grm ⊂ Aff. Find an element f ∈ Grm not in Rot. (b) Show that Rot is subgroup of Bij(R2 ). (c) Find two sets A, B ⊂ R2 so that A ∼Grm B but A is not congruent to B in the geometry (R2 , Rot). You need to prove why there is no f ∈ Rot so that f (A) = B. (3) Show that all parabolas are affine-congruent. Show that all hyperbolas are affinecongruent. (4) A diameter of an ellipse is a chord which passes through the origin. Prove the following Conjugate diameter theorem: Let ` be a diameter of an ellipse. Then there is another diameter m so that • the midpoints of all chords parallel to ` lie in m, • the midpoints of all chords parallel to m lie in `. 1
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