You are required to show your work on each problem on this exam. The following rules apply:

Transcription

1 Math 5335 Fall 2017 Eam 2 12/5/18 Time Limit: 80 Minutes Name (Print): KEY This eam contains 8 pages (including this cover page) and 13 problems heck to see if an pages are missing Enter all requested information on the top of this page and put our initials on the top of ever page in case the pages become separated You ma not use our books notes or an calculator on this eam You are required to show our work on each problem on this eam The following rules appl: If ou are appling a theorem ou should indicate this fact and eplain wh the theorem ma be applied Do not trivialize a problem If ou are asked to prove a theorem ou cannot just cite that theorem Organize our work in a reasonable tid and coherent wa Work that is disorganized and jumbled that lacks clear reasoning will receive little or no credit Unsupported answers will not receive full credit n answer must be supported b calculations eplanation and/or algebraic work to receive full credit Partial credit ma be given to wellargued incorrect answers as well If ou need more space use the back of the pages learl indicate when ou have done this Do not write in the table to the right Page Points Score Total: 100 You ma use the following results on the eam without defining or proving them The distance between points (a b) and (a d) in the Poincaré Half Plane is ln(d/b) The distance between points P 1 and P 2 on the line (!) with angles t 1 \(! + 0)(! 0)P 1 and t 2 \(! + 0)(! 0)P 2 is ln [(csc t 2 cot t 2 ) / (csc t 1 cot t 1 )]

2 HI ll Math 5335 Eam 2 Page 2 of 8 12/5/18 1 (8 points) Let D be a conve quadrilateral which is not necessaril a parallelogram trapezoid or other familiar shape Let W X Y and Z be the midpoints of the sides as shown in the generic diagram below Prove that WXYZ is a parallelogram w LI ti z LI W X Man approaches possible including : Wt Y If t tc TD ) X t Z L ( t tct D) Z Y Since WtYXtZ WXYZ is a H gram D 2 (6 points) Suppose a parallelogram D and triangle 4P are constructed on the same base and D and P! are all on the same line ` which is parallel to Prove the area of D is twice the area of 4P P D L ` ( man approaches possible ) h h h The 111g ram can be split into 20 s r a with same base I T ED b/c D is all gram ) 3 O s have same height h Thus HODIHHODHIHOPH HDH Hence HDH 2110 ell

3 bisector II lso f Math 5335 Eam 2 Page 3 of 8 12/5/18 3 (3 points) Let (1 1) (6 3) and (2 8) Find the centroid of 4 in rectangular coordinates G ( I s If t tc ) ( 912) ( 34 ) 4 (6 points) Prove that the perpendicular bisectors of (the sides of) 4 all intersect in a point J which is equidistant to all three vertices (You can use the diagram below for convenience but our argument must appl to all triangles) 1 No two sides of 0 are It their bisectors are not 11 hence and must intersect m k e Let h perp of T l n I m T Let J hnl prop of L bisectors JE k I FI I JI I J Jee IET f Isis I Hence IF so J equidistant to c In particular III I I Fcl JE m so all 3 I bisectors are concurrent at J 5 (6 points) Given 4 letd and E be the midpoints of and as shown Using an appropriate methods from the course prove DE k and DE 1 2 D ) t E Multiple approaches possible D ODE b SS Similarit t I s L LDE j corresponding angles DI HT OI D LI E E D II E D It and HE DH L It ll

4 2) Thus Hence Math 5335 Eam 2 Page 4 of 8 12/5/18 6 (6 points) Let D be a rhombus Prove the diagonals are perpendicular D r rhombus is a H gram so the diagonals Ill 6 I I I E bisect each other ll four D s are I congruent b SSS there are 4 congruent angles at E which combine to measure 2 it each of those angles is Is and the segments are I 7 (6 points) Let m {kxk 2} {( ) : } Find the coordinates of the reflection of each of the following points across m Graph the original points P Q and R and their reflections P 0 Q 0 and R 0 Note that the gridlines are drawn ever half unit in this picture P (0 1) Ilp ) Q D ( 94 ) (Poor planning m part that p on Q P are so close to each other µ p 14014/9 Q ( 3 4) 3414*1341 m Q P sorr I IQ) 4 HI / a I k) he R R R (0 2) Ufo 2) I O ( its on the minor )

5 Thus Math 5335 Eam 2 Page 5 of 8 12/5/18 8 In each part below sketch `0 the reflection (ie inversion) of ` across the mirror m lso give the equation for `0 (a) (5 points) m : and ` :( 1) m l " ` m and l intersect at l s ± Ez) so those pts are in l! l contains center of m El a line l is X z l (b) (5 points) m :( 2) and ` : 4 ` m n l ( 40 ) is fied so 14 ol El e El center ( 20 ) te (20) Ell so fl l a circle not a line I lso 14 2) El reflects to 1311 ) m ( 317d L (c) (5 points) m : and ` : ll points X t l have l There sent to points X s t m Hltll ll p Hence l is 2tZ4 cl

6 Inf In Note Math 5335 Eam 2 Page 6 of 8 12/5/18 On this page all points lines segments and distances are in the Poincaré Half Plane 9 Let (1 1) (1 2) and ( 3 1)! (a) (10 points) Sketch the (Poincaré) lines and! Find the equation for each line T : I 2 enter of radius is H T must be ( I ol f It nl/r5 to ) o T : ftpt75 (b) (3 points) Find the length of segment W/ formula on front : DI)?) 2 ( or Hutt ) 10 Let ` be the line directed b (2 0) and (1 0) (a) (5 points) Sketch ` on the grid below Then sketch and give an equation for a line m which contains the point ( 1 3) and is asmptoticall parallel to ` Haq 2 of od ( th (b) (3 points) Give an equation for a line k containing (3 1) which is ultra parallel to ` or eplain wh none eists ( Man answers possible) ( 4Pt22 (31) satisfies the egn and PDI s are tra o ) that > > 2

7 too IT HI Math 5335 Eam 2 Page 7 of 8 12/5/18 On this page all points lines segments triangles and areas are in the Poincaré Half Plane 11 (a) (5 points) The area of an tripl asmptotic triangle with Poincaré Direction Indicators P (p 0) Q (q 0) and R (1 0) as shown on the left is Use this fact to prove the area of the tripl asmptotic triangle 4 on the right is also P Q Let R ) HoneHHoaRHiHoeRH Hoaenll IT t IT IT IT (b) (5 points) The picture below shows a singl asmptotic triangle 4 formed b the lines and p 3 in the Poincaré Half Plane Find the area of the triangle using an valid method ftpbvert Ma If angles The 2 0 g Is II _ Is

8 Math 5335 Eam 2 Page 8 of 8 12/5/18 12 (5 points) Recall the incidence aioms from hapter 10: I1 For an two distinct points P and Q there eists a unique line that is incident with both P and Q I2 Ever line is incident with at least two points I3 There eist three points such that no line is incident with all three Define a 4 point geometr as follows The points are the ordered triples (0 0 0) (1 0 0) (0 1 0) and (0 0 1) s it happens those are the four vertices of a unit tetrahedron as shown in the picture below line is defined to be a set of three points which from the vertices of a triangle in the tetrahedron; for eample {(0 0 0) (1 0 0) (0 1 0)} is a line because those points are the vertices of the triangle on the bottom of the tetrahedron Does this 4 point geometr satisf the incidence aioms? Justif our answer Noe it doesnt satisf I 3 n 3 pts form a line 13 For each of the following statements indicate whether it is True or False b circling the corresponding answer riefl justif our answer (a) (4 points) Lines and µ in the Poincaré Half Plane can never be ultra parallel The alwas share PDI ( no o ) 0 True False (b) (4 points) onformal a nities preserve the area of quadrilaterals True 0 False onformal affinities scale which affects area