Intermedite Mth Cirles Wednesdy 17 Otoer 01 Geometry II: Side Lengths Lst week we disussed vrious ngle properties. As we progressed through the evening, we proved mny results. This week, we will look t vrious side length properties nd we will prove some results. Some of this mteril will e fmilir nd some of this will streth wht you lredy know. Prolems From Lst Week We will tke up four or five prolems from lst week. Complete solutions n e found on our wesite t http://www.em.uwterloo./events/mthirle presenttions.html. Getting Strted The Pythgoren Theorem: In right-ngled tringle, the hypotenuse is the longest side nd is loted opposite the 90 ngle. In ny right-ngled tringle, the squre of the hypotenuse equls the sum of the squres of the other two sides. In the tringle illustrted to the right, + =. Proofs of The Pythgoren Theorem: If you do n internet serh you will disover mny different proofs of the Pythgoren Theorem. If you go to the link http://www.ut-the-knot.org/pythgors/index.shtml#84, you will find 98 of the proofs grouped together. We will present three proofs here. Proof #1: The first proof presented ws visul proof. It will not e inluded in these notes. 1
Proof #: This proof ws not overed in the leture. Strting with the leftmost right tringle, rotte 90 to the right to rete the seond tringle. Rotte the seond tringle 90 to the right to rete the third tringle nd rotte the third tringle 90 to the right to rete the fourth tringle. This proess retes four ongruent right tringles. We will now reposition the four right tringles to rete the following figure. The figure is squre with sides of length. We n see tht the sides re eh length ut re the orners 90? Let the ngle etween side nd side e α. Then the ngle etween side nd side is 90 α. Eh orner then onsists of n α nd 90 α. The ngle t eh orner is α + 90 α = 90. The figure in the entre is squre. Eh side is units. The lrge squre is mde up of four ongruent tringles nd smller squre. We will onstrut n eqution using re. Are of Lrge Squre = Are of 4 tringles + Are of inner squre ( ) = 4 + ( ) ( ) = + ( + ) = + This is not the only wy to rrnge the tringles. A figure n e reted with lrge outer squre of side length + nd smller squre of side length inside long with the four tringles. This will e left s n exerise for the student to pursue.
Proof #3: This proof is ttriuted to Jmes Grfield, the twentieth President of the United Sttes. He silly tkes the first nd fourth tringles from our group of four tringles nd stks them on top of eh other s shown. At the point where the three tringles meet stright line is formed. Let the ngle etween side nd side e α. Then the ngle etween side nd side is 90 α. The remining ngle etween the two sides of length is 180 α (90 α) = 90. The lrge figure is trpezoid tht ontins three right ngled tringles. (The justifition tht the lrge figure is trpezoid is stright forwrd nd is not inluded here.) As in proof # we n form n re eqution. Are of Trpezoid = Are of Two ongruent tringles + Are of Isoseles tringle ( ) h ( + ) = + ( ) ( + ) ( + ) = + ( + )( + ) = +, fter multiplying through y + + = + + = A Pythgoren Triple is triple (,, ) of positive integers with + =. Wht Pythgoren Triples do you know? The following hrt illustrtes severl Pythgoren triples. The smllest side length is n odd numer. 3 4 5 5 1 13 7 4 5 9 40 41 11 60 61 Look for ptterns in the tle. For exmple, nd re onseutive integers, is even nd is odd. The sum + ppers to e perfet squre. Cn you predit the triple in whih the smllest numer is 13? Cn you predit formul for generting ny Pythgoren Triple with, the smllest numer, n odd numer 3. We n prove tht, for n n odd integer 3, then (n, n 1 proof will e left for the student., n +1 ) is Pythgoren Triple. This 3
If tringle hs two ngles equl, then the two opposite sides re equl. Tht is, the tringle is isoseles. The proof of this is left for the student. On the first night we proved the se ngle theorem for isoseles tringles tht sttes: if tringle hs two equl sides, then the two opposite ngles re equl. The ove sttement is lled onverse. When sttement nd its onverse re oth true, we n stte them together using if nd only if (IFF for short). The Isoseles T ringle T heorem n e stted: A tringle hs two equl sides IFF it hs two equl ngles. x x If A < B, then <. C If <, then A < B. If, nd re the side lengths of tringle, the Tringle Inequlity tells us tht + > nd + > nd + >. Cn you explin why this is true? B A There re two kinds of speil tringles. The first hs ngles 45, 45 nd 90. The seond hs ngles 30, 60 nd 90. If the shortest side in eh hs length 1, wht re the other side lengths? These n e sled y ny ftor. 45 30 45 60 4
Congruent Tringles Two tringles re lled ongruent if orresponding side lengths nd orresponding ngles re ll equl. In other words, the tringles re equl in ll respets. Sometimes, fewer thn these 6 equlities re neessry to estlish ongruene. Some wys to determine tht two tringles re ongruent: Side-Side-Side (SSS) Side-Angle-Side (SAS) Angle-Side-Angle (ASA) Right Angle-Hypotenuse-Side (RHS) One two tringles re proved to e ongruent, ll of the other orresponding equlities follow. Similr Tringles Two tringles re lled similr if orresponding ngles re equl. If two tringles re similr, then the orresponding pirs of sides re in onstnt rtio. In this exmple, A = X nd B = Y nd C = Z. Therefore, ABC XY Z. Then AB XY = AC XZ = BC. In other words, the tringles re Y Z sled models of eh other. Two tringles re lso similr if two pirs of orresponding sides re in onstnt rtio nd the ngles etween the sides re equl. One similrity is shown then the orresponding pirs of sides re in onstnt rtio. In this exmple, CB ZY = CA ZX = 1 nd C = Z.. 3 As result of similrity, B = Y, A = X nd BA Y X = 1 3. 5