Euclidean and Non Euclidean Geometry

Problem 11.2, 12.1 a, 12.1 b Due on March 14th

Problem 11.2

Three Points Determine an Isometry

To analyze what all isometries are, we need the following very important property of isometries:

Prove the following: On the plane, spheres, or hyperbolic planes, if f and g are isometries and A, B, C are three non-collinear points, such that f{A) = g{A), f(B) = g(B), and f{C) — g(C), then f and g are the same isometry, that is, f[X) = g(X) for every point X.

HINT:

Problem 12.1 a Dissect Plane Triangle and Parallelogram

a. Show that on the plane every triangle is equivalent by dissection to a parallelogram with the same base no matter which base of the triangle you pick.

HINT for 12.1 a:

Part a is straightforward, so don’t try anything complicated. You only must prove it for the plane — a proof for spheres and hyperbolic planes will come in a later problem after we find out what to use in place of parallelograms. Make paper models, and make sure your method works for all possible triangles with any side taken as the base. Make sure that your proof works for triangles whose heights.

are much longer than their bases. Also, you need to show that the resulting figure is a parallelogram.

Problem 12.1 b

b. Show that, if you assume AP, then on a plane every parallelogram is equivalent by dissection to a rectangle with the same base and height. Show equivalence by subtraction without assuming AP.

HINT for 12.1 b:

Problem 13.1 is due on March 20th

Please see Page number 179 of book for the suggestions. There is a complete detailed suggestion for the problem which if I copy will be long.

Moreover, I have attached the file with comments so that it will be easier for you to look for the mistakes. All last three assignments were graded as 77, 83, and 85 respectively. Assignment carries 40 percent so need to be figure out the way of not losing the points. Click on yellow box for all the pdf file with “graded ” letter on the last of filename to read the comments.