Prévia do material em texto
Scanned by CamScanner C ' D N S T R U C C IQ . N E S E N U k 1 . R C UN E E R 园 ? 协 - Æ a - Scanned by CamScanner B Scanned by CamScanner Scanned by CamScanner 114g J o s é 主 u i s M e z a B a r e e n a G e o r n e Ð l a PA EM . D QS CI RCUNFEFeENC1A$ r Bn r a d io [A B + B C - A c 2 A D IV U A B · C D N o TA E t ! Ei r n 1i n o ßg 1l ï a " i nscr i l a" s e r / i . . . . . Æ E z A P B = go û t Æ E . e = 10 J g « r a q u r? e s t á c o n t e n id a e n o r r a te s t ei d e i 1l r o c ie e Ma ) y \ e l 1 r m in o " c i r c u n s c r i 1o " e s c 1* a 1rd o Bma gur a c o n t i e n e O b s e r v a c i o n e s a a i r a (e s t ie c ì r / o r o c 1ea o l o h a c e c o n l e l t e r ) a . p o r d o s pu n to s c u a }e s q u ie r a p a s a n in f in it a s c lr c u n f e re n c ia s 魏 蕊蕊瓣 B * CA W à - - j b . P o r t re s p u n to s c u a Be s q u ie r a n o c o l En e a le s , . , P a s ? 0 0 7 m l Scanned by CamScanner G e o m e t r fa ï n g J o s é L u i s M e z a B a r c e n a E s a q u e l c u a d r i lá t e r o c u y o s v é r t ic e s p e r t e n e c e n a u n a m is m a c i r c u n f e r e n c ia C U A O R IM T F R O S FA BN S C R IPT IB L E GO N0 /ì Æ QNE S + P A RA OU E u N i : i X X S c r i ta , A B H e 3 ln s c r 1pti b lé C o n s e c u e n c ia + í j = 18 0 ° 2 S i : 0 · . A B C D e s In s c r ìp Bi b e d e l p r im e r te o r e 怀 旧 e n d o n d e u n 2 11 n g u Bo e x t . Rio ,d e l c u a d ri lá t e l o in s c Dto e s ig u a l A BC D e s a u n á n g u lo In s c r i & t bBe in te r io r E n t o d o c u a d r i lá t e r o : S i : O = P in s c r /t o s u s d ia g o n a Be s D E s a q u e l c u a d r i[á te ro c o n v e x o q u e p u e d e Æ in s c r it o H e a q u i l a m a te m át ic a , la cr eac i ón m á s o r HB ; n a l d e B /n g e n Bo h u m a n o (W n ite he a d ) 0 0 8 Scanned by CamScanner f n g 1 o s é L u is M e a B « r c e ï 1a G e o t r1e tr i t t Æ T O S N O T - J p u n t o s n o t a b le s d e u n t r iá n g u lo s o n a q u e l lo s p u n t o s d e t e r m i n a d o s p o r la i n te r s e c c ió n d e la s l i n e a s n o t a b le s d e u n a m is m a e s p e c ie (m e d ia n a s , b is e c t r ic e s , a l t u r a s . y m e d ïa t r ic e s ) U . . . M R IC E N T R O ( d e f l lyliñ a ® i+l c 0 s e @Æm P l e .+ p r o p i e d a d e s E s e l p u n t o de in te r s e c c ió n d e la s m e d i a n a s d e l t r ián g u lo , s e le d e n o m i n a t a m b ié n b a r ic e n t r o ö c e n t ro d e g r a d a d E n la fig u r a T ra z a m o s la ba s e E n e B tr ián gu lo A BC k ] m a m o s e l s e g Se o b s e r v a qu e lo s t r iàn gu k»s M G N y Io s la d o s de l r e s pe c t iv o t r ián gu lo ant i com¢»ment ar i o S l S t <e n e · . . . B ÀwnpAa1?=B?<. i F pa r a s 나 s re s p e c t iv o s t r ián g u lo s a n t ic o m p \e m e n tä r io s De t A A B C E B b a r ic e n t r o de to do t r iá n g u lo , e s t a m b ié n b a n c e n t r o pa r a s u r e s p e c tiv o tr ián gu lo a n l lc o m p <e m e n ta r io b l Æ 1 · E I s o r to c e n t r o d e to d«» t r iá n g u lo , e s c ir c u n c e n tr o d e i \ S i en u n tr iá n g u lo d o s m e d ia n a s s e c o r t a n pe r p e n d ic u la r m e n te , s e c u m p le q u e la d /s t a n c la de l b a r i c e n 0r o a l t e r c e r v e r 【ic e e s ig u a l a l la do o p u e s to A To de m t f ís ic a n o p a s a o e . i 1 a g e o m e t r ía (D e s c a r t e s ) 0 0 9 E ¢ m = 1 Scanned by CamScanner G e o · 1n e t r fa I t Bg . Io s è L t t i s M e z a Ba r c e n t t ダ )e 1PiIC E N T R O ( } j í pr i E · I a RÁF lc o l c A RACTSTI CA 名 To d o t r i n g u Bo t ie n e u n s ó lo in c e n t r o - _ ' IE D ニ ウ ( o 。 国 龟 C a d e u n i r lo s Es a q u e : q u e , p u n to s d e t a n ge n c ia d e la c i r c u n fe r e n c ia i n s c r i t a e n u n tr iâ n g u lo d a A M N P t r iá n gu lo S e c u +T \p le q u e e l e n * O d e u n t r iá n g u b e s e l ta n g e n c ia í d e l A A BC c ir c u n c e Dt r o d e u n t r ià n g u Bo l a n g e n c ia l M A ° 1" > in c e n t ro d e f A A B C Æ r c u n c e n t r o d e l 1H M P N T bd a m $ f ís ic a n o p a s a d e u n a g e o m e 0r a (D e s c a r :e s ) 0 1 0 I J < - 0 Z U 9 Z g Scanned by CamScanner Bt Bg . J o s é L t t i s M e z a B a r c e r 1a G e o BB1e t r f a i@ e x c e ist r o GA RA CS J CA S E l e x c e n t r o s ie m p r e e s tá e n e l e x t e , u f d e l t r iá n g u lo y e s te p u n to e s e q u /d is t a n te d e lo s t r e s la d o s d e t r iá n g u Bo To d o t r iá n g u lo t je n e tr e s e x c e n t r o s P a r a h a l la r u n o d e lo s t r e s e x c e n tr o s d e 나 n t r iá n g u lo b a s ta c o n t ra z a r 나 n a b is e c t r iz in te r io r y u n a b ts e c t r iz e x t e r io r (d e do s á n g u lo s d if e r e n te s ) 6 b ie n ba s ta ' c o n t r a z a r d o s b is e c t r ic e s e x te r io r e s Lo s e x c e n tr o s s o n c e n t r o s d e Ba s c ir c u n fe r e 1ì c ia s ' e x in s c r it a s a l t r iá n g u Bo c u y o s ra d io s s e d e n o m in a n EX R A D I OS I : Ex c e n t r o r e la t iv o a B C 0 11 Scanned by CamScanner Scanned by CamScanner 1n g j o s é L u js M e za B a r i e n n G e o n Be l r i a O RT O C E NT R O (H } DEÆN: c di i : F1cQ L PROPEDADES ( To d o t r iá n g u lo t ie n e u n s o lo o r t o c e n t r o p a r a h a lla r e l o r to c e n t r o d e u r l t r iá n g u ¢o b a s t a c o n t r a z a r la s E l - e n t r o n o s i e m p r e e s u n pu n t o i n te r /o r a u n t r iá n g u lo E n u n t r iá n g u lo a c u tá n g u 1o e l o r to c e n t r o e s u n p u n to in \e r it»r E n u n t r ià n g u lo r e c t á n g u Bo e l o r to c e n t l o s e h a lta e n l v r tîc e E n u n t r iá n g u lo o b tu s á n g u Bo e B o r to c e n t r o e s u n p u n to T- 了剽 AE" UL O A C U T IÓ N G U § - rLGG E C T ÁN C U L O l nl Ár 4GuLo o m u し a A ' ' u I B= - í i i i m l = a NGULOI C- - G R ÁF IC O E s a q u e l t r iá n g u lo ' 1 E l o r t o c e n t r o d e u n tr iâ n g u lo a c u t á n g u Bo - 1 c uy o s v é r t ic e s s o n c e n t r o de s u t r iá n g u lo O R T IC O lo s p ie s d e Fa s al 2 E n to d o t r iá n g u lo a c u tá n g u lo s u s la d o s s o n b i s r a s de u n l r ián gu t r ic e s e x te r io r e s d e d ic h o l r iá n g u lo P E DA L lo , a l c u a l s e le de 3 L o s v e r tic e s d e u n t r iá n g u Bo s o n lo s e x c e n t r o s d e n o m i n a t r iâ n g u k ) d ic h o t r i n g 나匕 P EDA L OR T I CO ' 4 . n t od o t r ián g u b ob t us á n g u l o, l os l ad o s qu e m a n e l m u lo o btu s o y Ba a lt u r a r e la t iv a a l te r c e r L î PA C la d o , s o n b is e c t r ic e s in t e r io r e s d e l r e s p e c t iv o tr i O A M N P Æ th O rt ic o o P e d a l g u lo O RT 1C O 8 A A Bc + 1h A n t lá r t ic o 5 F n to do t r iá n g u lo o b tu s á n g u \o e 1 la d o q u e s e w e i F a B á n gu lo o M u s o c o n la s a Hu r a s r e la t iv a s a lo s o t r o s " H ° + I n c e n tr o d e l ' M N p y 1o s v é r tìc e s A , B y C s o n ' A H y C s o n e x c e n t r o s d e l t r iá n g u fo pe d a l j e x c e n t r o s d e l A p e d a l · B e s e l o r to c e n t r o d e l A A H C E r s u n d o e s tá c a d a v e z m à s do m Bn a d o p o t Ba m e e r r á t lc a (A F R a m b a u d ) 0 13 j 1 Scanned by CamScanner Scanned by CamScanner +n g . J o s é L u isM e z a B a r c e n a G e o Bn e ¢r f« j (P r o p De a lte r n o s in te r n o s ) d o n d e : H + O r t o c e n t r o d e l I 1 A B C B G > Ba r ic e n t r o d e l A A B C ; / a ' " " . ' " ' inis ) ? H 0 15 ダ Scanned by CamScanner Scanned by CamScanner f 11 g . J o s é h Bis M e z t ï B a r c e ï 1 a G e o n t e f r f t B 1B O S IC IO N ES DE L A R E C TA D E E UL E R S EG ÚN EL 1I PO D E T R IA d Rr ÁNCu I s E C U M P L E Q U E j C O ND iC i ó ° X u . ' " t " , : G B a r ic e n t r o T T E Ex c e n t r o re la t i I A ( e m á s n i ta s r e c ta s de E u le r E l o r t o c e n t ro , e l b a ri H > o r t o c e n t r o : c e n t r o y e l c ir c u n c e n G b a r ic e n t r oo t ro s e e n c u e n t r a n e n ' u n a l:n e a r e c ta (r e c ta O > c ir c u n c e n t r o d e E u le n m 4 § ' " ° (o b tu s o ) / & ° To d a m i f ís ic a n o pa s a d e . s a g e o m e t r ía (D e s c a r t e s ) 0 1 7 ー コ 9 Z く の コ 1 国 0 Scanned by CamScanner G e o n Be f r fa í B1g J o s é L r* i s M e z a B t Br c e r t a T E O R E M A T E O R E MA D E L A C IR C U N F E R E N C IA D E E U L E R L o s p 나 n t o s m e d io s d e lo s la d o s , lo s p ie s d e la s at r a s y lo s p u n to s m e d io s d e lo s s e g m e n t o s c p r e n d id o s e n t r e c a d a v é r t ic e y e l o r to c e n t ro d e u n t r iá n g u lo , s o n n u e v e p u n t o s q u e s e e n c u e n t r a n e n u n a m is m a c i r c u n f e re n c ia A e s t a c i r c u n f e r e n c ia s e Be d e n o m in a " Ci r cunf er enca d e 1o s n u e v e p u n t o s " e q u i lá te r o s e h a l la n e n l in e a r e c ta F E l r a d io d e la c ir c u n f e r e n c ia d e E u le r e s ig u a l a p u n t o m e d io d e l s e g m e n to q u e t ie n e e l H F - F O d a d o H F = F O , d o n d e F p u n t o d e F e u e r b a c h A le r F 4 C e n t ro d e la c i r c u n fe r e n c ia d e lo s n u e v e pu n to s E l p u n to d e F E U E R B A C H (F ) e s e ï c ir c u n c e n t r o Eu Be r ia n o s d e l t r iá n g u b m e d ia n o o c o r n p 1e m e n t a r ìo 0 1 8 Scanned by CamScanner f t r g . 1 o s è L t t is M e z a B t i r c e n a G e o n i e f r i a T E O R E M A D E L A R E C TA D E E U L E R T E O R E M A D E L A R E C TA D E S IM P S O N n u e v e p u n t o s p u n to d e F e 나 e r b a c h ) , e l b a r ic e n t r o y L o s p ie s d e la s pe r p e n d ic u la r e s b a j a d a s d e s d e e t c i r c u n c e n t r o r e p r e s e n t a n a Fin e a d o s e n e s e o r u n p u n t o a r b i t r a r io d e u n a c i r c u p í r e n c ia a ]o s d e n , a c u y a r e c ta s e s o p o r te s e 1e d e n o m in a " R e c ta la d o s d e u n t r iá n g u lo in s c r it o e n . . a ú lt im a s e d e E u le r . E n c u e n t r a n e n u n a m is m a lm e a r e c ta H + O r to c e n t ro N : s o n fu e n te s e o ïin e a :e s a 0 1 9 Scanned by CamScanner G e o m e t r i a 1n g . 1 o s è L t t is M e z a B a r c e n a S e o b s e r v a q u e 1o s c u a d r iBá t e r o s L P M A y P M N B s im p s o n 0 2 0 A S Scanned by CamScanner í B1g . . Ro s é L u is M e z a B a r c e n a G e u n i e f r í tB v per r «»_t ! acn S e a ' ) ç ' e l á n g u lo fo r m a d t » p o r m y n , r e c t a s d e S im ps o n , D e m o s t r a r e m o s q u e P a r a e Blo , s e p r o c e d e c o m o s ig u e T E O R E M A a la c ir c u n f e r e n c ia c i r c u n s c r it a e n " W D e m o s t r a r La a lt u r a B M d e u n t r iá n g u lo A B C , c o r ta a A C e n M y q u e A C e s m e d ia t r iz d e l s e g m e n t o H ; ' (H e s e l or c e n t r o ) Lu e g o s a C 0 2 1 Scanned by CamScanner G e o m e t r fa +BBg . J o s é L t t i s M e z a U a r c e n a T E O R E M A I D e ( i ) y (2 ) s e d e d s o n ig ; U c e q u e lo s á n g u lo s K P S y N S p D E M O S T R A C IÓN P r o lo n g a m o s la a ltu ra B Q ha s t a " K , s e t ie n e q u e PK c o r ta a A C e n " R " , p e r o A C e s la m e d ía t r iz d e H K CTe o r e m a a n t e r io r) , e n to n c e s e l t r iá n g u lo H R K e s is Ms c e le s y c o 1T H» B K e s p a ra le la a P S s e t ie n e Ta m b ié n m · LB KP = m q K H R - m < K P S . . . . (1 ) m B K P = m B C P = m N S P . . . . . . . (2 ) S e o b s e ï 园 只 丐 î A h o r a o b s e r v a m o s e m t l iá n g u b P H R ; dE e s 0 2 2 Scanned by CamScanner ï n g J o s é l u i s M e z a b a r c e n BB G e o m e t r í t t L A C I R C U N F E R E N C IA D E L O S N U EV E Pa P U N T O S Y L A S R E C TA S D E S I M P S O N Po r t a n t o T T ' e s ta m iu d d e Bd lâ m e t r o P P ' , p ) r : o t a n t o r r ' L a c ir c u n fe r e n c ia d e lo s n 나 e v e p u n t o s e s tá fo r m a d o s e r á e l d iá m e t r o d e ta c * C U n · e n c /a d e to s n u e v e p u n o s : ! ' ' ' " 0 2 3 飞 Scanned by CamScanner 024 Scanned by CamScanner h , g . M s d L u is M , z · B " " . . a Ge " " , I. C R I T E R I O S D E C ON ST R U C C I ÓN - / Hi b - B U SC A N D O c U A D R I L ÁT E R O S I N SC R bPT . B L E S " Ø S E 5 i s e t ie n e a s ig u ie n te f ig u r a : s i s e Db . e rv a la si g' " nt ' "g' ' " o n la s 5l gui e' " . N ' " ac t ' " s t ' " as De he c h o . E+ e " i to r e & e s e n ta e l 1% d e u n tr ab a jo qu e ' " Ui 【a d d 99% d B a qu e lfo q u e s e l la . a fra c a s o 0 2 5 Scanned by CamScanner Ge o 1BBe t r i t t f n g . J o s é L Bt / s M e z a B a r c e r1a ÇA Q I I I A q u i s e o b t ie n e Bo s ig u ie n te S i s e t ie n e la s ig u ie n t e f ig u r a c o n t a le s c a r a c t e r is t ic a s a 冒目 迅 - - e a ' 1 u à . V - A qu í s e üe n e 8 6 ° 3 » e ' " ° 3 \ 1 ( 1 S U G E R E N C I A A q u í s ó Bo he p re s e n ta d o t re s c a s D s e m p le a n d o t r a z o s a u x i l ia r e s p a r a e n c o n t r a r c u ä d r i lát e r o s in s c r ip t ib le s ; lu e g o e n b s p r o b le m a s r e s u e lt o s e l e s t u d ia n t e le c to r e n c o n t r a r d ife r e n t e s t ip o s d e t ra z o s a u x i l ia r e s . p u e s e x is t e g r a n v a r ie d a d d e e Mo s \ De h e c h o , e B e x it o r e p re s e n t a e l 1% d e u n t r a b a j o o u e r e s u lt e d e l 9 9% d e a q u e =o q u e s e Uar na í r a c a s o 0 26 A \ Scanned by CamScanner Scanned by CamScanner G e o n Be f r l « 卧 圈圈 C a lc u la r " x " 1n g 1o s é L u i s M e za B a r c e r 1a i p a s o N ° 3 : s o b s e r v a q u e s e o bü e n e u n c u a d n lá \ te r o .n s c r ip t ib te R e s o :u c ió n P Þ く P a s o N ° 2 : A Plic a n d o la s p r o p ie d a d e s d e la m e A q u i s e o b t ie n e , d e la f ig t r r a , u t i liz a n d o Ba p r o p 0 d ia t r iz 0 2 8 r 1 Scanned by CamScanner Scanned by CamScanner G e o m e t r i a 1ï 1 B 1 o s é L u i s M e a B a r c e n a ' - « T R A Z O S A U X I L - S E M P L E A N D O L O S P U N T O S N O T A B 1 E S " C A S O A : E M P L E A N D O E L C I R C U N C E N T R O " L u e g o s e o b s e r v a e n la f Bg u r a L S I s e t ie n e la s 0g u ie n te f ig u r a c o n t a le s c a r a c te r ís t ic a s e - > A y c a g u d o s S a b e m o s q u e t o d o t r i i n g u Bo , e s tá c o n t e n id o e n A p l ic a n d o la s p r o p ie d a d e s q u e s e c u m p la n e n la c ir c u n f e r e n c ia , . , L T Lu e g o , e n la f igu r a r e s u lt a n te s e - p le t a r à lo s á n g u lo s in te r n o s , pa r a lu e g o h a tia rla s o )u c iö n d e l p r o ble m a To d a e d u c a c ió n c ie n t íf c a m j e ) s e ' n c l a c D N ta m a te m à t c a e s l mpeec t a e n s u b a s e (A u g u s t o C o m te l 0 3 0 B Scanned by CamScanner S e o b s e r v a e n la f ig u r a ; u n t r i n g u lo q u e U P o r á n g u lo in s c r it o » x z 3 0 ° T o a a e d u c a c ió n c ie n t if ic a q u e n o s e m 1c 0a c o n Ja m a t e m M Þc a e s m p e r fe c ta e n s u b a s e : Agl 5s o C o m e t 0 3 1 i Scanned by CamScanner Ge o Bme ï r i a BBBg 1o s é L u i s M e z a J3« r c e n a " EJEM P LO N ° o 1 I P a s o N ° 3 : C o m p le ta n d o á n g u lo s e n :a f ig u r a . . ' . ' P a s o N e Z A h o r a a p »c a m o s Ba s p r o p ie d a d e s q u e it o s L u : . 、 、 ン ダ l o p a s o N o 4 s e o b s e r v a fn t e m a m e n t e e n e t g r â f ic o u n t r iá n g u lo js o s c e le s ヒ ノ L a m a l e m ic a o o e s u n a c ie n c ia s m o :a c ie n c ia {F e f Fx A v e r b a c h ) 0 3 2 む 妇 Scanned by CamScanner . . . Scanned by CamScanner G e " " 1, Ï. ï " g . Jo s L u i s M e z° B a rc . . là,+£å".=t=·. . I à , En e l p ro b re m a Pa s o N ° 3 E n la f ,g ' ' " e s u .ta n te s e o b s e rv a L . ° 2 - 5 0 - - C 034 Scanned by CamScanner Scanned by CamScanner Ce " "u , f a l n g . J o s é L . l s M e z · B " " " p a s o N · 3 0 b t e n e m o s ra s t r e s b+s e c t r +c e s d e r ' C a lc u l . r Y Re s o l u c ió n Pa s o N ° 1 S e o b s e r v a h is R r . t r i, e n u n . . n j . n t o de l tdá n gu b , ' . , ì " . _ _ - " P a s o N ° 4 A h o ra o bs e r v a m o s U n . a d r i t e r o , d o p a s o N0 2 Redi zamo. el si gui e' " Ua z o p a ra o b t d ' ' " p l, [a p r o p je d a d de l ° p a n t . ló n o r e bo t e " n e r M r a bïs e c t r iz . - A s í o b t e . E m o s e l ín c e n l , o e n e t t r iaBn g u lo s o mb r e d 口 I n c e n t . . Þ - ー ト悬乙 X - 3 5 · 0 36 Scanned by CamScanner Scanned by CamScanner G, O . . Ef r l , / , g . . Ro sé L u i s M em B . . c , n . C B 0 C : E M p L E A N D O E L E X C E N T RO . n m b . Se ile n e . - - Se o b uB . . Ia b t re s bl. . d , i. . Se . B t . Ia s Ue s bls e c u]c e s Y s e c u m pl° A. ' " ' " 】 ! . . A EC = J 03 8 Scanned by CamScanner b rg . J o s d L u is Jrl , Z° B . . . a . . G. o m e t l 1h P s o N · 3 0 b te n e m o s la ' " Uï ent e f jg u ra d o n de s e c u 呷 le Re s o l u c ió n ' " ' " s o e æ n g u r a : la s â , g . 1o · in te r n o s y / 一 熟曳 V z s>. X - 135 · Pa s o N ' 2 o bs e rv a m o s l ' "gul ent epropi . d. d 8 ís e c tr iz E - Ex c e n l r o 0 3 9 Scanned by CamScanner Scanned by CamScanner I n g . J o s é L Bt i s M e z a B a r c e t t a G e o m e f r k 0 9 C 13 C E A 10 B 1 8 C H 5 A 10 ° B 15 ° C2 0 o D ) 2 0 ° E ) 18 ° D ) 2 2 ° 3 0 ' E ) 3 0 0 10 C a l c u la r x 14 C a lc u la r x 国 A ) 1 5 ° B) 2 0 ° C ) 2 2 ° 3 0 ' A ) 3 0 ° B ) 3 6 ° C ) 2 2 ° 3 0 ' D ) 2 5 ° E) 3 0 ° D ) 2 0 o 3 0 E ) 4 5 ° 1 1 C a ic u 1a r x , 15 C a lc u la r x 12 C a lc u la r x 16 C a }c u la r x A ) 32 ° B ) 3 0 ° C ) 64 ° A ) r r B) 3 0 ° a C ) 6 0 ° 2 09 D ) 5 8 ° E ) 4 8 ° D ) 2a E ) 3 a 0 4 1 Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner - - 一 一 女 、 一 、 、 、 一 一 一 一 一 Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner r l t g . 1 o s é L u is M e z a ß a r c e Bt a G e o m e t r ía ry 7 1 C a lc u la r x 园 四国 八) 18 ° B ) 2 4 o C ) 3 0 ° Æ) 3 6 ° E ) 4 5 o A ) 3 B ) 3 6 ° C ) 4 5 o n ) 6 D° E ) 5 0p 7 5 C a lc u la r x . 7 6 I 八) 2 a B ) 3 0 a C ) 6 0 2 a D ) 6 0 o E ) 3 0 o 7 3 C a lc u la r x 3 0 ° B ) 2 5 o C ) 3 5 ° Ð ) 2 0 ° E ) 4 5 ° 7 7 C a lc u la r x A 1 0 0 ° B H 2 0 ° C 3 5 o A 1 0 o B H 2 ° C 4 ° D 1 5 0 ° E H 6 0 o D 1 5 o E H 6 o 04 9 Scanned by CamScanner Scanned by CamScanner 1n g J o s é L 1t i s M e z a B a r c e n a G e o Bt +e t r fa 86 C a lc u la r x 9 0 C a lc u la r a A 12 5 c H e D ) 2 0 ° E ) 2 4 ° A 1 0 B H 2 C H 4 q 1 5 E H 8 A ) 3 0 ° B ) 4 0 ° C ) 4 5 ° D ) 5 0 ° E ) 5 5 ° D ) 5 5 ° E j 3 5 ° 9 2 C a lc u la r x 8 8 C a A ) 2 4 ° B ) 22 ° 30 ' C ) 2 6 ° D ) 3 0 o E ) 36 ° 9 3 C a lc u la r x 8 9 C a lc u la r x X A 1 0 B 12 C H 4 A 6 0 ° B 7 5 ° C70° D 1 5 E H 8 ° D 9 0 ° E 8 0 ° 05 1 Scanned by CamScanner Scanned by CamScanner j , g - Jß s d L H j s rw , z . B n " " . G, m . Err ' " 1 ' t i : =h¯ p t . oßì m i *H P a s o N ° 2 . 0 b s e r ve mo s l a si g' " ' " l ig' " E s u n c u a d r ilàt e r o 〒 n s . . Ip t ib le d o n d e s e c u m p le , , : , x tì3 Pa s o N ° 3 . Tã m b ién s e c u m p le 9 : 053 Pa s Scanned by CamScanner Scanned by CamScanner ー ー 一 一 ー Scanned by CamScanner Ge o r r t e t r l a 11t g J o s é L u i « M e z a B a r c e n a · / I " Pa s o N ° 3 S e o b s e r v a u n c u a d r i lá te r o in s c r l pt 0b le A q u í s e o b s e r v a d o n d e c u m p le la p r o p ie d a d d e l p a n t a tö n r bl ema o 6 c a lc u la r " x " ー じ 0 5 6 : j Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner / I IG. J os é L I , l r M ez · B . r c u m G " " e » / . 々以拉一 一 一 \ A \ P " " ° 5 S . o b s e r v a u n . . A d r rlâ t . . Do n d e c u L - Pa s o N · 3 . Se t ie n e u n a re c ta m e d l a t r iz d o n d e s e I · Pa s o N ° 4 . A ho ra o b s e rv e m o s la s ig " " t e f ig " " 0 5 9 - Scanned by CamScanner Scanned by CamScanner 1n g . J o s é L u is M e z a B a r c e n a G e o n r e l r f r r { 1 p a s o N * 3 C o n s t r u im o s u n t r iá n g u lo e q u i lá t e r o d e P r o b l e m a 11 . C a lc u la r x la s Bg u le n te m a n e r a : R e s o l u c i ó n g P a s o N ° 1 F o r m a m o s u n t r iá n g u lo ìs á s c e le s e x te r n a m e n t e 0 6 1 \ Scanned by CamScanner G e o m e f r ta f n g J o s é L u i s M e z a B a r c e n a S e c u 0 6 2 Scanned by CamScanner 11t g J o s é L u i s M e z a B a r c e t1a G e o n re t r i a p a s o N ° 7 E n la f ig u Fa s e t ie n e u n c u a d r ilá t e ro Pa s o N ° 4 C o n s t r u im o s u n t r iá n g u lo e q u ìlá te ro d e d o n d e s e c u m p le P a s o N ° S S e o b s e r v a la s ig u ie n t e f ig u r a H a l la n d o p o r s u p le m e n t o P a s o N ° 8 A ho r a o b s e r v e m o s e l t r a pe z o id e s im t r ic o f o r m a d o a l in ic io 0 6 3 : 0 ° Scanned by CamScanner G e o n w f r Ba ï n g . J o s é L u i s M e z a B a r c e n a A q u i s e c u m p le 3 0 " + O + x + 3 0 " O = 9 0 ° 6 0 ° Æ x = 9 0 ° X = 3 0 ? P r ob l ema 13 . a l cu B ar x P a s o N ° 4 C o n s t r u im o s u n t r iâ n g u :o e q u ìBá t e r o d e la s ig u ie n te m a n E r a 0 64 r Scanned by CamScanner p a s o N ° 7 . En la f ig u r a s e o b " - " " u a d ' " t er ° 0 6 5 Scanned by CamScanner G e o m e t r i a f t Bg . Jo s é L BBis M e z a B a r c e Bt a 0 6 6 Scanned by CamScanner ï Bt g . Jo s é BUs f e a B a r c e n « G e o n 1e f r l a í \ P a s o N ° 6 S e o b t ie n e u n c u a d r i:á te ro d o n d e s e R e s o l u c i «i n c u m p le la p r o p ie d a d d e l " p a n t a ló n o r e b o t e " >( = 2 0 ° P r o b l e m a 1 4 C a lc u la r x P a s o N ° 2 A h o r a t e n e m o s e n Ba f ig u r a 0 6 7 Scanned by CamScanner G e o n ï e t r í a f n g 1 o s é L u i s M ez a B ar c e r 1t B / p a s o N ° 3 s e o b s e rv a u n c u a d r ilá : e r o d o n e c u m p a s o N ° 2 s e Tie n e u n a re c t a m e d ia t r iz d o n d e s e ' p le Ba p r o p ie d ád d e l " p a n ta Bó n o r e b o t e " : c u m p le P r o b l e r h a i 5 C a lc u la r a S e c o n s t r u y e u n t r iá n g u lo is ó s c e le s @ ol uci d P a s o N ° 1 S e o bs e r v a u n á n g u lo d e 3 0 ° e n t o n c e s p a s o N ° 3 . S e o b t ie n e u n c u a d r i lá t e r o d o n d e s e f or ma mo s e x t e ri o r m e n te u n t r ián g u lo e q u i låÈe r o d e c u m p le la p r o p ie d a d de l p a n t a ló n o de l re b o te la s ig u ìe n te m a n e r a 06 8 苎 Scanned by CamScanner . Scanned by CamScanner Scanned by CamScanner 11t g . J o s é L u is M ez a B a r c e 11a G e o n 1e f r l a P a s o N ° 2 C o n s t r u im o s u n t r iá n g u lo is ó s c e le s P a s o N ° 5 S e t ie n e u n a r e c t a m e d ìa t r iz e n e J P a s o N ° 4 C o n s t r u im o s e x t e r n a m e n t e u n 0 7 1 Scanned by CamScanner Ge o n , d , h , g . J OsÉ L u i s M , Z° ß . . c , " ' p a s o N ° 7 . Se o b tie n e u n cuadr át er o d o n d e - Ca s o g e n e r a 1. Ca [c u M r x . c u m p le la p ' " i ' " d d e l pa n ta lón o re b o te 立 Pa s o N ° 1 Re a hz a m o s e l s ig ' " n t e t ra z o ; Pa s o N ° 2 E n la fìg . Ra s e t ie n e 回国 S . g ún e 1 P · . B l 0 7 2 Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner : n g . J o s é L u i s M e z a ß a r c e n a G e o m e t r i a , ' g u io is ó s c e le s d e ¢ 飞 P 0 7 5 Scanned by CamScanner G e e m e f r i a 歹 t Bg J o s é L u i s M e t t i B a r c e ï 1a p a s o N ° 5 A h o r a s e t ie n e d o s t r i n g u lo s R e s o l u c i ó n 1 c o n g r u e n t e s C a s o (L A L ) / ' ' ' ' ' ' ' ' ' ' ' ' ' . Ó P a s o N ° 1 S e o b s e r v a u n á n g u lo d e 3 0 ° e n la P a s o N ° 2 C o n s t r u ir n o s e x t e r n a m e n t e u n tr i å n 1o e q u i lát e r o P r o b l e m a 2 1 C a lc u la r x 3 0 ° + a 12 0 ° 2 0 7 6 Scanned by CamScanner h , g . J . sè L u is M ez . 8 . Rc è. . · 专 四国国匡国 c a 【c u ra · x p a s o N5 . " ' " n ' : Y, . · 0 7 7 Scanned by CamScanner G e o n 1e t r fa . n g . J o s é L u i s M ez a B a r e e n a / . * b t d o s t r i ' " u1os c o n g r u e n / I e . / i X B £ " ' " 07 8 Scanned by CamScanner r ï t g 4 o s é L t t is M e z a Ba r c e r t a P r o b l e m a 2 3 . C a lc u la r x o 1u c i ó n i Pa s o N ° 1 C o m p Be t a m o s lo s á n g u lo s i nt er nos d e l p a s o N ° 4 H a c e m o s e l s ig u ie n t e t r a z o p a r a fo r m a rt ri á n g u lo : u n t r iá n g u lo e q u i lá te r o y a s í o b te n e m o s d o s t r iá n g u lo s c o n g r u e n te s e , X P a s o N ' S S e o b t ie n e d o s t r iá n g u lo s c o n g r u e n te s , L . 0 7 9 Scanned by CamScanner ' ' Scanned by CamScanner / 1' g . M sé L . Is M e t ° B . . c , n . Ge o . . a , 1tr f i e J ' td 0 8 1 Scanned by CamScanner G e o 1 n e t r Ta l n g J o s é し u i s M e za B a r c e 1ta ' 7 p a s o N ° 5 s e o b \ îe n e u n t r iá n g u lo e q u /Bà t e r o p r o b l e m a 2 5 c a lc u la r x \ P a s o N ° 1 C o m p le t a n d o Bo s á n g u !o u n t e r n o s p a s o N ° 2 Re a i z a mo s e l s ig u ie n te t r a z o p a r a t e r 0 8 2 Scanned by CamScanner r " . J o s é L . is M , z · B o , c e " " C . . M , M i . 0 8 3 Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner - - Scanned by CamScanner G e o n i e ¢r f a 1n g . 1 o s é L u i s M e z a B a r t ¬ f +a P a s o N ° S A h o r a s e tie n e u n c u a d r i lát e r o do n d e P a s o N ° 1 C o m p le ta n d o á n g u lo s e x t e r n o s c u m p le [a p r o p ie d a d d e l p a n t a 1ô n o r e b o t e 5 . B0 1 ° . : : : is Ms c e 1e s , OBB Scanned by CamScanner Scanned by CamScanner G e t n BBe f r i t t I n = . Io s é し u is M e e n B a r c e BBa P a s o N ° 2 C o m o s e t ie e u n t r iá n g u lo is ó s c e 1e s , t r a z a m o s s u a Jt u r a , d o n d e c u m p le - - 82 ° . . P a s o N ° 3 Ta m b ié n s e t ie n e u n a m e d ia t r iz d o n d e c u m p l i rá : b 1e m a 3 1 C a s o g e n e ra l Ca lc u la r x 8 2 ° 34 ° 3 4 ° v a s o N - 4 A n o r a s e o b s e r v a u n c u a d r i là t e r o d o n d e c u m p le l a p r o p ie d a d de l p a n t a ló n o re b o te / 0 9 0 Y Scanned by CamScanner Scanned by CamScanner C · Scanned by CamScanner Scanned by CamScanner G e o m e f r 1a 1n g . J o s é L u i s M e z a ß a r c e n a / 09 4 ニー ニ . - Scanned by CamScanner / 1 : - M s L n b M ° B . . . En , Ge o . . Et r l Q Pa s o W 3 . Fo rm a m o s u n tn á . g . I° is ós c e le s d e la s ig ' " n t e m a n e , a ' 回国国目国 c a lc u la / A £3 Ó' " l " , e n to n c e s e m p le " " m s p ro p+ e d 7 - Pa . . N ° 1 C ° ' " ' " n d o lo s á ' " I' " nt er nos d e la f i g' " P a s o N ° 6 . Se o bt te n e " " . d , ifáte r o d o n de c u / To ° Y p le ia p r o p ie da d de t p a n ta lön o re bóte - c ' / ï \ - Pa s o N ° 2 . 0 b s e rv a m o s e n u n á ng u lo e x te r io r , te n e r bis e ct riz e x t e r io r 09 5 Scanned by CamScanner ' " _ . = . , . 9 Scanned by CamScanner - Pa s o N ° 5 E n d ic h o cuadmát or ocumpl e Pa s o W 2 Tä m b ién s e o b s e r . a u n a b +s e c t r iz i n t m + P a s o N ° 3 . O bte n e m o s la . Ig . Ie n t e fig " " o n lo s s Pa s o N ° 4 Se o b s e ' " a . Ig u +e n te f ig " " u n tr i á . . x - 3 g u lo is ó s c e le s - C a so g e n e ra l , C a lc u la r x Pa s o N ° 1 0 b s e p / a m o s la . Ig . Ie n te f ig · Y . - 2a - Scanned by CamScanner ó e o M Bb 1n g , J o s é L u i s M ez ° B . r e e n ° - X P a s o N ° 6 . 0 b s e rv a n d o 1' "g" " / ] \ , &x ' " a A q u : o b s e rv a m o s 团团醒醒黜 o 9 a Scanned by CamScanner ー ノ ー! i Scanned by CamScanner C , O- l r i . I . g . . Ios t L . i s A1ez a B ° , c e n ° Scanned by CamScanner r Æ喀 . J o s é L t t i s M e l o B a r c e r t a G e o n Be t r i a R e s o l u c i ó n P a s o N ° 7 Te n e m o s la sgui ent e f ig u r a P a s o N ° 2 S e o b t ie n e u n c u a d r i lâ t e rc d o n d e ' s 10 1 Scanned by CamScanner Ge o m e t r i a 1n g . J o s é L u i s M e z a ß a r c e ï Ba T a m b ie n c 나 m p le し 、 P a s o N ° 4 A q u i s e c u m p leaa / 1 0 2 Scanned by CamScanner In g . Jm l L , JS W « 1 ß " " ' " G " " r r i . - Ca lc u la r a . - P o N ° 1 . S e c o m p le ta lo . án g . :o [ i n mo s e n la fìg ' " - Pa s o W 4 . A ho ra t ra z a m o s hl m e d ia t r iz d o l 10 3 Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScannerScanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner Scanned by CamScanner