Ptgoren Teorem nd Trigonometr Te Ptgoren Teorem is nient, well-known, nd importnt. It s lrge numer of different proofs, inluding one disovered merin President Jmes. Grfield. Te we site ttp://www.ut-te-knot.org/ptgors/inde.stml will provide ou wit over 50 different proofs, nd links to oter soures. Mn proofs of te Ptgoren Teorem (inluding Grfield s) involve res, wi we ve not studied et. Our proof insted involves similrit: Ptgoren Teorem: In n rigt tringle wit legs of lengt nd nd potenuse of lengt,. + ~ Given ª wit rigt ngle t, let,, nd. rop perpendiulr from to side nd ll te foot of te perpendiulr. Finll, let,, nd. Note tt ª is rigt tringle tt sres p., ª is similr to ª. Similrl,ª is rigt nd sres p, so gin, ª is similr to ª. + + ( + ) Using rtios of tese similr tringles, we n estlis tt nd. Tese give us 2 2 tt nd. dding,. Note: Te ove set-up of similr rigt tringles n now e used to prove severl oter results,
inluding: 2 or ( is te geometri men of nd ) nd Just for fun: Grfield s proof. Strt wit rigt tringle wit legs of lengt nd, nd potenuse of lengt. Etend te side of lengt to lengt +, nd onstrut perpendiulr of lengt. Form segments s sown elow. Now te re of te trpezoid formed n e lulted in two different ws: s trpezoid, te re is ½ te sum of te ses, times te 1 1 2 2 1 2 2 eigt:. On te oter nd, ( + )( + ) ( + 2 + ) ( + ) + we n lulte te re of te lf squre of lengt nd dd it to te sum of te two tringles: 1 2 1 1 1 2 1 2 2 1 2. Tus, or. + + + 2 ( + ) + + 2 2 + (I et u n t do tt. He ws ild left eind, I guess. )
Teorem (onverse of te Ptgoren Teorem): If + ten p is rigt ngle. ΔEF Δ is tringle su tt ~ rete tringle wit F, FE, nd pfe rigt ngle. Ten ΔEF + ( E) in te Ptgoren Teorem pplies nd. ut we know tt in Δ, 2 + 2 2. So E, nd te two tringles re ongruent SSS. Ten p pfe PF, nd so is rigt ngle.
Trigonometr: In wt follows, we dopt te onvention tt in ª,,, nd. Tus is te lengt of te side opposite verte, et. We dopt te usul definitions: in rigt tringle ª wit rigt ngle t we define : sin μ( ) s μ( ) os μ( ) se μ( ) tn μ( ) ot μ( ) We usull revite sin µ(p) s sin, depending on ontet to mke it ler tt we re tking te sine of te mesure of te ngle t verte. For onveniene in working wit otuse ngles in tringles, we define: We lso define: sin 0 0 os0 1 sin 90 1 os 90 0 sin180 0 os180 1 sin(180 μ( )) sin os(180 μ( )) os Finll, we note tt te usul si identities follow esil from tese definitions nd te 2 2 Ptgoren Teorem (e.g. ). sin + os 1 We ll prove te Lw of Sines, nd one se of te Lw of osines. Rell te onvention tt in ª,,, nd. Tus is te lengt of te side opposite verte, et.
sin sin sin Lw of Sines: For n tringle ª,. sin sin ~ If we n sow tt for n oie of nd in our tringle, te oter equlit will follow repeting te proof for nd eiter or. Te proof ten redues to ses depending on te reltive size of te ngles t,, nd. se 1: Te ngles t nd re ot ute. rop perpendiulr from to sin ; sin te perpendiulr ten flls on segment. Ten nd we ve sin sin. Te foot of
se 2: Te ngle t is otuse (te se for eing otuse is etl nlogous). rop te perpendiulr from to Ten we ve: ; in tis se, te foot of te perpendiulr is su tt **. sin nd sin sin( ) sin(180 μ( )) sin( ) sin sin. Ten se 3: ngle (or ) is rigt ngle. sin sin sin 90 1 sin 1 sin Ten, nd, so we ve.
+ 2os Lw of osines: For n tringle ª,. ~ One gin, tere re tree ses, nd I m going to do te es one: Note tt 2 2 + ; + ; ( + ) ;os. Ten: 2 os ( ) ( ) 2 ( ) 2 2 + + + + + + + + + 2 ( ) 2 2 ( )( ) + 2 + 2 + 2 2 2 +