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# FNDNS GEOMETRY I MATH 5200

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This 14 page Class Notes was uploaded by Joanne Bergnaum on Saturday September 12, 2015. The Class Notes belongs to MATH 5200 at University of Georgia taught by Staff in Fall. Since its upload, it has received 97 views. For similar materials see /class/202082/math-5200-university-of-georgia in Mathematics (M) at University of Georgia.

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Date Created: 09/12/15

Math 52007200 Geometry Class Notes Note to class is my symbol for parallel N for similar Wednesdax1 September 3 2008 Laura Hill explains homework question 1 McC Shows hidden she starts with segment AB She did it right since as points A and B move the Segment AB is still divided into 7 equal parts You see she did it right since points move and stay equal Laura Use properties of lines Made segment AB put point P not on AB constructed segment BP plotted point X on BP constructed circle centered at X thru point B made 5 more congruent circles along segment BP The last point on segment BP was connected to A Then lines were constructed thru all the points on BP that are to the line thru point A Reminder To make tool select your diagram go to the Tool menu the double arrows at the bottom of the tool bar create new McC s suggestion Instead of making segment BP make a ray Ifyou put point to close to point P on segment BP you may not have enough space to mark off the 7 congruent segments Also to get the rst point on the ray instead of just putting the point there make a circle centered at B and construct the intersection of that circle and the ray Do you understand Why does it work What properties of lines did you use Laura n is 7 so you make n 7 l circles connect last point and make the lines Similar angles Michelle adds corresponding angles McC All segments you made are congruent if all here congruent how are they congruent on other side Laura shows examples of N triangles there are 7ido you see them all Here s an important theorem If parallel lines are cut by transversals those transversals divide the lines proportionally Adam I didn t use lines at all McC Yes there s another way We look at Adam s picture which I don t have access to so any attempt of mine to explain it will be lost sorry I can t do it justice Adam but it was very cool Here s what I caught in class Midpoint ofthe given segment made squares the size of half the given segment that gives a l by 2 rectangle then things with diagonals this is where it was hard to followitwo diagonals with rectangles diagonals of squares their intersection gives third and a perpendicular dropped from that point to the original segment marks off a third of the segment McC In doing 17 constructs harmonic series something we should remember from 3200Not as easy to see why this works This is too interesting so it will take a lot of class time more than we have to explain The idea is that it s an inductive procedure third to fourth to fth to sixth to seventh However if you play with GSP enough you can stumble upon this Russ presents homework question 7 Area ABCA 163 in2 Given AABC construct a second tr1angle so that the 39 2 ratio of areas is 2 move the points and the ratio stays Area ABGF 33926 In the same and the angles stay the same Area ABGF 200 similarity ratio is 141 Area ABCA 39 B A F mACAB 54190 szCA 6150o szFB 54190 szGF 61500 Show all hidden Russ Similarity ratio is 141 found that playing which gsp so area ratio would be 2 That 141 is root 2 71 know from seeing it all the time Made lst triangle then constructed circle with radius AC then right triangle AEBD to get hypotenuse EDiknew I needed hypotenuse not sure why Questions from class How dial you get the larger circle Using ED as radius and B as center What is special aboutED About that right triangle It s isosceles So what The hypotenuse is root 2 times leg then larger circle is center a with that hypotenuse as radius How alialyoufinalpoint F Ray from B to A where ray intersects bigger triangle other leg of triangle must be to third leg of original How dialyou have point b How alialyou know where BC is That s given Ray AC where larger circle interested triangle 1 got this statement form class but looking at the picture it doesn t make sense McC Technically speaking point on object isn t a construction available to us but I mnot making a big deal This is especially nice drawing because but this way the auxiliary triangle isn t overlapping the original That similar triangle can be anywhere and this is the most convenient way to do it constructing new on top of old This is very important b c emphasizes the importance of area of similarity Another big theorem If length ratio is r then area ratio is r2 so if area ratio is s then length ratio is root xT McC This is key example in course area is key and sadly neglected in school mathematics volume also Charnelle explains 3 Start with circle point P Segment OP Then midpoint of OP call it M circle with M as center thru P and 0 Select intersections of two circles Draw perpendicular to those points to get tangent line C Yes perpendicular went thru point p McC What s snother way to connect point to P and you know its perp Why did this work How did you know this would give you tangent Chamelle Any line perpendicular to radius is tangent to 4 x the circle McC How do you know the line is perpendicular When you constructed perpendicular how did you know it would go thru point H Chamelle These segments are equidistant to circle McC How did you know it was right triangle How is hypotenuse related to the circle Chamelle Its diameter McC We have theorem If a triangle inscribe in a circle has one side as the diameter then the angle opposite it is right angleinot the last time you ll see this one of his favorite theorems This is good way to make right tri given hypotenuse Central angle is twice the inscribed angleiso we know this theorem as special case So if writing proof you can use the original theorem or this corollary McC Weak point is that many people didn t explain why their construction worked There s not any big difference only a difference of degree in explaining why a construction works and proving won t make a big distinction between these two at this point in the class Explain why works based on other stuff you already knowiproblems are constructed so that you can use what we ve already done in this course Proofs not in specific form just very clear explanation using facts we already discussed or you already know McC Golden rectangleiwho knows what it is Chamelle Diagonal of rectangle to side is l to 53 it involves golden ratio and diagonal McC Not diagonal sides Base to height is golden ratio but what is golden ratio Ancient Greeks had properties of this rectangle involving proportions Cut a square off what s left is another golden rectangle has same proportions as what you started with In algebra which the Greeks didn t have but we do lucky us Height is x length is x y How can we say large and small have same ratio as long to little side x x xy i pretty to do construction repeatedlyiway to V y x construct logarithmic spiral Some algebra phi p say fee is golden ratio symbol z x x lx x li lt0 02 ol 02 p l0 1i12 411 21 lix14 135 QZ zi 2 2 1J 0 2 Since we re dealing with distance we only need positive McC way we construct uses diagonal of something so maybe this is what Chamelle remembers l l not1ce interesting a 1 7 7 1s 0618 0 0 Greeks used this ratio in their architecture popular term project lots of examples of the golden ratio in nature too Wikipedia good resourceone guy who updates math Wikipediaithat s why they are good Greeks could construct this golden ratioi but we ll draw since time limited Start with short side and create long side Construct 1 root 5 over two using Pythagorean theorem Start with short side construct square construct midpoint of one side connect midpoint to vertex square to create right triangle lx2x2h2 x 4 2x2h2 1x 4 2 q 2 The hypotenuse is the length we need Construct a circle that hypotenuse as radius and center as vertex of right angle of right triangle Then our pink segment is the golden ratio Notes on F139iday1 September 5 2008 Remember to check the website for notes and such Let s discuss next week Here are graduate students pick groups 20 minute presentation We are making the transition from constructions to proof How you present is your decision better if not grad students not everyone has to present need a write up for this Email les by class time General discussion of all projects I groiectishow how to do all the other constructions midpoint parallel line perpendicular line angle bisector circle by center and radius on the construct menuiusing only what the Greeks had a available intersection lineraysegment circle by center and point 239quot1 Qroiect 7Construct a regular pentagon use fact that the ratio of diagonal to an edge is the golden ratio to do this construction extra credit is to gure how why diagonal to edge is the golden ratioiyou won t have time to present both of these 33911 and 4th Qmiects iProperties of Quadrilaterals what are the de nitionsiuse only most basic properties form which other properties can be derived De nition should be re ected in the word used to name the object parallelogram quadrilateral with opposite sides parallel theses two groups are connected so they will need to communicate Rectangle is quadrilateral where adjacent sides are perpendicular Rhombus is quadrilateral with all four sides equal Square is a regular quadrilateral Kite has 2 pairs of adjacent congruent sidesquotDifferent Math 52007200 Notes to supplement class presentation 13 October 2008 This was not a homework review day saved for 15 OCT but will do SAS similarity proof TERMINOLOGY REVIEW related to proportion Consider 2 segments each divided into 2 pieces What does it mean that they re divided in the same proportion proportionally this says slw X x we have seen this with the theorems about diagonals oftrapezoids and with similar triangles such as when using SAS similarity lfthese segments are two legs of similar triangles for example we have c z and we can say i i a x Since c ab and z Xy we can substitute and find 2 m31 1xigz a a x b we need to be careful and be sure to compare corresponding pieces Some information on the proof We are going to start with pieces of same segments and extend this to sides of triangles Remember what we have to work with from the axioms and basic theorem groups all but similarity theorems and Pythagoras We can use isosceles triangles parallels congruence and area We went through some ofthe work on the Euclid s Elements page Book VI Prop 2 Dr McCrory has posted the proof and I am not going to go through that Dr McCrory also referred to this particular proposition as the Proportional Size Theorem and pointed out a few times that the results are based on use of area In particular the fact that the area of a triangle is independent ofwhich base you chooseuse when calculating area INFORMATION FOR EXAM ON 22 October Wednesday will be a review ofthe week s homework There will be a new assignment due Friday the last before the exam There will be practice problems over the weekend Be careful when working outlined proofs and supplying reasons steps may involve more than one axiom or theorem The work is done We just need to explain it ie provide the reasons for the steps of the proof 15 October 2008 Homework 12 review Since last time posted on the website Proof of SAS Proof of Straight Angle Theorem Existence of parallels in Michelle s notes And all the materials from the basic theorems is now available online Considering question 1 The line joining the midpoints ofthe nonparallel sides ofa trapezoid is parallel to the parallel sides and equal in length to half their sum I assume the trapezoid is not a parallelogram What happens if it is a parallelogram K N O L 3 Outline of proof Drop perpendiculars from the upper verticesto the base and from the midpoints ofthe legs to the base Use Angle Angle to prove AKE similar to ANC and the same on the other side of the trapezoid Use the similarity ratio of21 We will have a rectangle MNOP and can work with the facts about rectangles We also have rectangle EKLF and can again use what we know about rectanglesto ultimately nd parallel lines and show that EF is parallel to CD and consequently AB We went back and used the proof showing that opposite sides of parallelograms are parallel After a lot of work we showed that we use the diagonals of EKFL and work again with properties of rectangles which nishes up the parallel part of the proof We nally looked at the relations from the similar triangles along with rectangle information to show that KN and OL are each of the base oftheir respective triangles and we added each of these to NO from the lower base that is congruent to CD Doing the algebra gave us that the midsegment is the average of the bases of a trapezoid Mary Catherine did a much simpler proof by constructing the diagonalso fthe trapezoid and working with the midpoints ofthe sides of the trapezoid She looked at the trapezoid at a time shoed that the segment using the midsegment created 2 similar triangles by SAS proving that the midsegment was parallel to the base and setting up the proportion to nd the length of the entire midsegment when she put the 2 halves of the trapezoid back together It used the existence of parallel theorem and took much less work than the earlier proof By going back to rst causes and the axioms it led to a much simpler proof using only parallel lines and SAS similarity MORE COMMENTS FROM DR McCRORY A list of useful theorems we can use will be posted On the exam he won t be as picky as on the homework we won t be restricted to axioms The most important thing is making logically correct arguments based on what we know and have proved in class You need to be able to state exactly what theorem you are using not just by problem 3 A Criterion for simple proofs they use fewer construction lines Math 52007200 Geometry Friday October 24 2008 Review of problem 1 from Test 2 Most people made the same mistake we need to learn from this MISTAKE Assume that every angle bisector is a perpendicular bisector of the opposite side The hypothesis actually says that the two points the incenter and circumcenter are the same the hypothesis says nothing about lines It s important to go back to the de nition not the construction Incenter is equidistant from the side of the triangle Circumcenter is equidistant from the vertices of the triangle To find distance from incenter t0 the side we must drop a perpendicular from the incenter to the side Brief Outline of proof We have six right triangles use Pythagorean theorem to show that all three sides the same then the sides of the original triangle are the same McC Thought this was an easy problem because I didn t foresee that people would make this false assumption Eric Can you use 306090 triangles McC That is circular reasoningiwould have to prove that its equilateral to get that it is 306090 triangles McC Purpose of this Can you assume things that aren t given give something familiar to see if you can step back and not make assumptions beyond what was given However this was more extreme than I had intended Solutions will be posted on the website Problem 2 for test 2 is interesting but not any tricks TRIG Emphasize relationship of trig to geometry there is a gap between geometry and trig in schools We cover some theory followed by some applications Detour from syllabus to introduction to nonEuclidean geometry Guest lecture from next semester professor Link on the website to definition of trig functions Only focusing on sine and cosine Best to know these before moving to others The definition is based on AngleAngle Triangle Similarity using internal ratios instead of going between similar triangles Sine and Cosine only depend on the measure of the angle What if you have other angles not the acute angles of a right triangle To deal with this use the definitions provided in calculus coursei essentially making coordinate system Be sure to keep track of positive and negative angles Bunch of great properties follow from these two definitions and they are included on the webpage Michelle Do we need to prove these properties McC Since these are direct consequences of the de nition you can just use them You should prove them for yourselves I will not prove them in class It will take some organization to see how this works Trig was developed after Euclidean Geometry because it involves measurement which was not considered by Greeks Came out of applications to astronomy For example 360 degrees corresponds almost to number of days in the year why 360 instead of 365 because 360 has lots of divisorsiastronomy isn t totally exactifor example leap years 7 possible that a year used to be defined as 360 days Homework for Monday The three most important properties of sine and cosine beyond the basic ones Standard notation Side opposite angle A is side a 1 Area half the product of two sides of the triangle and the sine of their included angle This is generalization of special case for right triangles Think about 2 cases with acute and obtuse angles 2 Law of Cosines The square of one side of atriangle is equal to the difference of the sum of the squares of the other two sides and twice the product of those two sides and the cosine of their included angle Again 2 cases to consider but this one is tougher 3 Law of Sines In a triangle the ratios of the length of a side to the sine of the angle opposite that side are constant 2 cases again Homework lsee the web We re focusing on Standard applications trying to give ideas for teaching how to use trig functions to simplify proofs in geometry Angle addition formulas coming next week Monday1 October 27 2008 Lind presents 1 Area at a Triangle Adam Did she go backwards She showed the formula 12 ab sinC is equal to area formula not derive it the other way McC This is a good formula You don t actually the construction line that gives the diagonal distance in practice So there could be a lake or mountain or something in the middle that keeps you from being able to measure this distance Laura presents 2 Law at Cosine Case 1 ltC is obtuse Must use property Cosine an angle is the opposite of the cosine of the supplement Adam Can you do this in one case McC No we can t do this in one case Its important that cos changes from being negative to positive Knowing what you are trying to get will help you figure out how to structure the proof you know you need cosine C so what is that in this diagram and how does it relate to the other things you need Marv Catherine presents 3 Law of Sines Her proof covers both acute and obtuse case Allyson Why McC Because height is same if the angle is acute or obtuse Our goal here is to see things from fresh viewpoint so we can close the gap between geometry and trig Some might say sine of angle and its supplement are the same and use that the supplement s sine is hb This falls back on properties It follows from our definition that sine of obtuse angle is hb you can also prove the properties listed on the definition page and use these Looking at definition Using similar triangles can scale any triangle down so that hypotenuse is l and use unit circle the de ning cosine as x coordinate of intersection of terminal ray of angle where initial ray of angle is positive xaxis and unit circle How can we prove from this de nition that angle and its supplement have the same sine Figure out way to get supplement with initial ray on pos xaxis Intersection of unit circle with line that does thru initial point that is to xaxis now one angle is supplement of the other then use given de nition to show that sine of angle and its supplementary angle are the same You can use the same to show the corresponding property for cosine Cosine of angle and cosine of its supplement are opposite Similar construction to show properties for negative angles Sin of angle is opposite of sine of negative angle Cosine of angle is equal to cosine ofnegative angle How can you construct the complement Re ect over y x Thanks to Adam and his gre studying This actual construction looks pretty complicated the upshot is sine of an angle is the cosine of the angle complement Pedagogical point from McC I hate trig because none of the formulas were ever explained You just have to learn them why aren t the formulas explained because of the disconnect between trig and geometry Other applications 1 Relation between area formula and law of sines Useful to realize that sin area formula is three different fomulas Set these equal and low and behold you get law of sines How lovely 2 Congruence theorems and trig formulas SAS congruence practical way to think of this if you given me an angle included and two lengths that uniquely determines the triangle The opposite side is interesting what formula gives you thisilaw of cosines shows exactly how the opposite side is determined from SAS How could I get the other two angles Law of sines Main thing you used to learn in trig soving trinalges Ran out oftime to discuss ASA and SSS Wednesday1 October 29 2008 Go over homework questions due next Monday 1 Website for compass bearing which has a lot of other fun stuff in it Always north or south then degrees to the east or west Yes southeast is 45 degrees east of south Compass reading is good application for the elementary kids thought maybe today s kids need GPS applications 2 tenth of a degree is tiny pretty sure you can t measure more accurately than a tenth of degree 3 use law of cosines 4 no comments MATH 52007200 Notes Dr McCrory Dr McCrory began class by saying that we will investigate spherical geometry this week He said that he likes to think of Euclidean Geometry as the mother and hyperbolic and spherical geometry as two sisters They can both be understood by their comparisons Dr McCrory said unfortunately spherical geometry is not taught much any more even though it is considered the NonEuclidean Geometry Spherical geometry is much older than hyperbolic and sometimes called Elliptic Geometry It was rst discovered by astrometry ancient Greeks and Babylonians However we do not have much time for the history of spherical geometry in only a week so let us begin Dr McCrory says that Geometry of the Sphere is a GSP software for spherical geometry However it attens out the spherical geometry and makes it dif cult to understand So we will not use the GSP program We will start with points A point in Spherical Geometry is any point on the sphere We may use a sphere of any radius but conventionally we will use a radius of 1 In Spherical Geometry lines and straight is an odd concept for Spherical Geometry the shortest path on a sphere are called geodesics Dr McCrory showed an example of a geodesic on the globe from a ight from Atlanta to Tokyo Flight goes directly over Anchorage Alaska Dr McCrory So geometrically how do we describe these geodesics How would you do a 3D construction Consider the concept of lines in Spherical Geometry 39 Dr McCrory said think about the center of the sphere 39 Dr McCrory make a suggestion to go back to the example airplanes take routes that correspond to the Great Circle routes the Equator There are an in nite number of great circles Antipodes the exact opposite point from a given point The great circle has the same center as the sphere Think of planes of great circles with a plane through the center of sphere Geodesics are arcs of these great circles Unlike Euclidean Geometry there may be more than one shortest distance especially if points are antipodes Geodesics play the role of lines A line is not in nite it is a circle 0 There are no parallel lines for any two great circles intersect at two points as long as they are not the same line 0 So we don t have the parallel axiom Next topic Triangles 39 One difference is that you can make a triangle with three right angles 0 Try to Visualize taking an apple and cutting it into eighths then you have eight congruent sections of an apple the face the skin of the apple is a triangle and all three angles are 90 degrees 0 Dr McCrory makes a 909090 equilateral or isosceles triangle on the globe All three sides are a quarter of the circumference of the sphere 39 Mary Catherine asks are all equilateral triangles 909090 39 Dr McCrory No It depends on how big the triangle is Visualization shows an approximation where the angles look like 60 but it is a little bit larger The smaller the geodesic triangle the angles approach 60 degrees and mirror Euclidean Geometry The 909090 is not the largest triangle you can make There are theorems that are true in Euclidean that are true in Spherical and many that are not 39 The most interesting one is the theorem about area of triangles The sum of the angles in a triangle will always be greater than 180 degrees 0 Allison asks is there an upper limit Think about there is not just one arc between two points 0 Dr McCrory I don t know if there is a maximum you can get Well there is a theoretical maximum So what should be the relation between the sum of the angles and the area in Spherical Here it is the efficiency or angle excess The sum of the angles of the triangle minus 180 Remember Hyperbolic is 180 minus sum of the angles and it is called the deficiency I Before we start the proof everything will be in radians The area of a geodesic triangle is the sum of the angles minus 71 Area of geodesic triangle Sum of the Angles H To do this proof understand the intersection of two great circles The region inbetween two great circles is called a lune A polygon with two sides Kind of like a slice of an orange A lune has only two sides and two angles and the angles are equal 0 0 Proof What is the formula for area of a lune with an angle a

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