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UCR MATH 133 - Examination 2 Mathematics 2007

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NAME:Mathematics 133, Fall 2007, Examination 2Answer Key11. [30 points] (a) State Pasch’s Posulate (also known as Pasch’s Theorem).(b) State the conditions that and ordered list of four coplanar points A, B, C, Dmust satisfy in order to be the vertices of a convex quadrilateral.(c) Explain why the ordered list of points A, B, C, D will give the vertices of aconvex quadrilateral if AB||CD and AD||BC.SOLUTION(a) This is given in the notes.(b) The points A and B must lie on the same side of CD, the points C and D mustlie on the same side of AB, the points A and D must lie on the same side of BC, and thepoints B and C must lie on the same side of AD.(c) If AB||CD, then A and B are on the same side of CD, and likewise C and D areon the same side of AB. Similarly, if AD||BC, then A and D are on the same side of BC,and likewise B and C are on the same side of AD.22. [25 points] Suppose we are given two triangles ∆ ABC and ∆ DEF such that∆ ABC∼=∆ DEF , and let G and H be the midpoints of [BC] and [EF ] respectively.Prove that ∆ GAC∼=∆ HDF and ∆ GAB∼=∆ HDE.SOLUTIONThis is given in the solutions to the exercises for Section III.2.33. [25 points] By definition and a theorem from the notes, the circumcenter V of∆ ABC is the unique point satisfying |V −A|2= |V −B|2= |V −C|2, and the circumradiusis the common value for the distance from V to these vertices. Find the circumcenter ofthe triangle in R2with vertices (0, 0), (3, 4) and (6, 0).10 POINTS EXTRA CREDIT. Find the circumradius of the triangle, and show it isa rational number by computing its value explicitly.SOLUTIONIn coordinates, the equations defining the circumradius are x2+ y2= (x − 3)2+ (y −4)2= (x − 6)2+ y2. If we subtract x2+ y2from all sides of this equation we obtain thesystem of linear equations 0 = 25 − 6x − 8y = 36 − 12x. Solving these, we obtain x = 3and y = 7/8.Extra credit question. The circumradius is equal to the distance from the circum-center to the vertices. For computational purposes, in this problem it is most convenientto pick the vertex (0, 0), for in this case the circumradius simply turns out to be the lengthof V . Thus the circumradius is given bys32+782=r242+ 7282=r25282and hence it is equal to 25/8 = 318.44. [20 points] Suppose that we are given an isosceles triangle ∆ ABC such thatd(A, B) = d(A, C) > d(B, C). Determine which of the following is true, and state thetheorem which yields this conclusion.|6ABC| < 60◦< |6BAC| , |6ABC| > 60◦> |6BAC|SOLUTIONSince the longer side is opposite the larger angle, it follows that |6ABC| >


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