Intermediate Math Circles Wednesday 15 October 2014 Geometry II: Side Lengths
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1 Intermedite Mth Cirles Wednesdy 15 Otoer 2014 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 Let us tke up three prolems from lst week. We will hve volunteers present solution for prize. Complete solutions n e found on our wesite t presenttions.html. Remrk from Lst Week s Lesson Our proof tht the interior ngles of n n-gon sum to 180 (n 2) only works if eh interior ngle is less thn 180. We ll these types of polygons:. 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 you will find 98 of the proofs grouped together. We will present two proofs here nd third one will presented online in the ompleted note for tonight s session. Proof #1: The first proof presented is visul proof. 1
2 Proof #2: In the note tht will e pulished online, seond proof is presented. Proof #3: Strting with the leftmost right tringle, rotte 90 to the right to rete the seond tringle. This proof is ttriuted to Jmes Grfield, the twentieth President of the United Sttes. 2
3 A Pythgoren Triple is triple (,, ) of positive integers with = 2. Wht Pythgoren Triples do you know? The hrt illustrtes severl Pythgoren triples. The smllest side length is n odd numer Look for ptterns in the tle. 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. Proof: 3
4 If tringle hs two ngles equl, then the two opposite sides re equl. Tht is, the tringle is isoseles. The Isoseles Tringle Theorem x x Relting Angles nd Sides C Tringle Inequlity Lw B A If, nd re the side lengths of tringle, the Tringle Inequlity tells us tht + > nd + > nd + >. Cn you explin why this is true? Speil Tringles 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.)
5 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. Four wys to determine tht two tringles re ongruent: 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. Given: A = X nd B = Y nd C = Z. Therefore, In other words, the tringles re 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 remining pir of orresponding sides re in onstnt rtio nd the other orresponding ngles re equl. Show tht the tringles re similr. 5
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