Download Midterm 2016 - b

Math 130
Midterm
Jamie Conway
Name:
Do not turn over this test until instructed to do so.
Write your name on all sheets you hand in.
Explain your work carefully for full credit.
You may not use any notes or textbook for this midterm.
Good luck!
1. (a) (5 pts) Let A, B, and C be distinct non-collinear points in the plane. Using a ruler and
compass, and knowing only how to find midpoints and draw perpendiculars, construct
the circle that passes through A, B, and C.
(b) (5 pts) Prove that your construction actually gives you the circle you claim. You may
use any grade-school level geometry, e.g. congruence theorems or symmetry arguments.
2. (5 pts) Let α = tan(1°). Is α constructible by straightedge and compass? Why or why
not? You may use any result we proved in class if you state it clearly, and you may use any
grade-school trigonometric identity.
3. (5 pts) Show that the axioms (I1), (I2), and (I3) are independent of each other. That is,
construct a model for a geometry that satisfies (I1) and (I2), but not (I3) (you’ll need three
models, one for each combination).
4. (10 pts) You are working in a model for geometry that satisfies Hilbert’s axioms (I1-3) and
(B1-4).
Let ]BAC be an angle, and let D be a point in the interior of the angle. Show that the ray
−−→
AD must intersect the segment BC at a point F . Cite any axioms you use; you may use the
fact that a line separates the plane into two disjoint sets. Hint: First construct a point E
such that C ∗ A ∗ E.
Axioms of a Hilbert Plane
Incidence Axioms:
(I1) For any two distinct points A and B, there exists a unique line l containing A and B.
(I2) Every line contains at least two points.
(I3) There exist three non-collinear points.
Betweenness Axioms:
(B1) If A ∗ B ∗ C, then A, B, and C are three distinct points on a line, and also C ∗ B ∗ A.
(B2) For any two distinct points A and B, there exists a point C with A ∗ B ∗ C.
(B3) Given three distinct points on a line, one and only one of them is between the other
two.
(B4) (Pasch) Let A, B, and C be three non-collinear points, and let l be a line not containing
any of A, B, or C. If l contains a point D lying between A and B, then it must also contain
either a point lying between A and C or a point lying between B and C, but not both.
Congruence Axioms:
−−→
−−→
(C1) Given a line segment AB and a ray CX, there exists a unique point D on CX such
that AB ∼
= CD.
(C2) The relation ∼
= on segments is an equivalence relation.
(C3) (Addition) Given three points A, B, and C satisfying A ∗ B ∗ C, and three points D,
E, and F satisfying D ∗ E ∗ F , if AB ∼
= DE and BC ∼
= EF , then AC ∼
= DF .
−−→
−−→
−−→
(C4) Given an angle ]BAC and a ray DG, there exists a unique pair of rays DE and DF
on either side of the line DG such that ]BAC ∼
= ]EDG ∼
= ]GDF .
(C5) The relation ∼
= on angles is an equivalence relation.
(C6) (SAS) Given triangles ∆ABC and ∆DEF , suppose that AB ∼
= DE, AC ∼
= DF , and
∼
]BAC = ]EDF . Then the two triangles are congruent.