DISSERTAÇÃO DE MESTRADO Uma Matemática Voltada para o Professor do Ensino Médio

  Mestrado Profissional em Matemática em Rede Nacional PROFMAT DISSERTAđấO DE MESTRADO

  Uma Matemática Voltada para o Professor do Ensino Médio Geraldo Ferreira Barbosa Filho

  ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❆▲❆●❖❆❙ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

  ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ●❊❘❆▲❉❖ ❋❊❘❘❊■❘❆ ❇❆❘❇❖❙❆ ❋■▲❍❖

  

❯▼❆ ▼❆❚❊▼➪❚■❈❆ ❱❖▲❚❆❉❆ P❆❘❆ ❖ P❘❖❋❊❙❙❖❘ ❉❖

❊◆❙■◆❖ ▼➱❉■❖

  ▼❆❈❊■Ó ✷✵✶✺

  ❯◆■❱❊❘❙■❉❆❉❊ ❋❊❉❊❘❆▲ ❉❊ ❆▲❆●❖❆❙ ■◆❙❚■❚❯❚❖ ❉❊ ▼❆❚❊▼➪❚■❈❆

  ▼❊❙❚❘❆❉❖ P❘❖❋■❙❙■❖◆❆▲ ❊▼ ▼❆❚❊▼➪❚■❈❆ ❊▼ ❘❊❉❊ ◆❆❈■❖◆❆▲ ●❊❘❆▲❉❖ ❋❊❘❘❊■❘❆ ❇❆❘❇❖❙❆ ❋■▲❍❖

  

❯▼❆ ▼❆❚❊▼➪❚■❈❆ ❱❖▲❚❆❉❆ P❆❘❆ ❖ P❘❖❋❊❙❙❖❘ ❉❖

❊◆❙■◆❖ ▼➱❉■❖

  ❚r❛❜❛❧❤♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ Pr♦❣r❛♠❛ ❞❡ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ ❞♦ ■♥st✐t✉t♦ ❞❡ ▼❛t❡♠át✐❝❛ ❞❛ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❡ ❆❧❛❣♦❛s ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ ❣r❛✉ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳ ❖r✐❡♥t❛❞♦r✿ Pr♦❢✳ ❉r✳ ❏♦sé ❈❛r❧♦s ❞❡ ▲✐♠❛

  ▼❆❈❊■Ó ✷✵✶✺ Catalogação na fonte Universidade Federal de Alagoas Biblioteca Central Divisão de Tratamento Técnico

  Bibliotecário Responsável: Valter dos Santos Andrade B238m Barbosa Filho, Geraldo Ferreira.

  

Uma matemática voltada para o professor do ensino médio / Geraldo Ferreira

Barbosa Filho. – 2015. 168 f. : il. Orientador: José Carlos de Almeida Lima. Dissertação (Mestrado Profissional em Matemática) – Universidade Federal de Alagoas. Instituto de Matemática. Programa de Pós Graduação de Mestrado Profissional em Matemática em Rede Nacional. Maceió, 2015. Bibliografia: f. 167. Índice: f. 168.

  CDU: 51:377.8

  ✐✐✐ ➚ ♠✐♥❤❛ ❡s♣♦s❛ ❏❛♥❝✐♥❡✐❞❡✱ ❝♦♠♣❛♥❤❡✐r❛ ♣❛r❛ t♦❞♦ ♦ s❡♠♣r❡ ❡ ❧✉③ q✉❡ ❣✉✐❛ ♦ ♠❡✉ ❝❛♠✐♥❤♦✱ ❛♦s ♠❡✉s ✜❧❤♦s ❱✐❝t♦r ❡ ▼❛②❛r❛✱ ✐♥s♣✐✲ r❛çõ❡s ❞❛ ♠✐♥❤❛ ✈✐❞❛✱ à ♠✐♥❤❛ s♦❣r❛ ❚❡r❡③✐♥❤❛✱ ❝❛r✐♥❤♦s❛♠❡♥t❡ ❝❤❛♠❛❞❛ ❞❡ ◆❡♥♦✱ à ♠✐♥❤❛ ❝✉♥❤❛❞❛ ❏❡❛♥❡✱ q✉❡ ❝♦♥s✐❞❡r♦ ❝♦♠♦ ♠✐♥❤❛ ✐r♠ã✱ s❡✉ ❡s♣♦s♦✱ ❊✉❝❧✐❞❡s✱ ❛♦s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦✱ q✉❡ ♥❛s ❤♦r❛s ♠❛✐s ❞✐❢í❝❡✐s ❡stá✈❛♠♦s ❥✉♥t♦s ♣❛r❛ ❛♣♦✐❛r ✉♠ ❛♦ ♦✉tr♦✱ ❛♦s ♣r♦❢❡ss♦r❡s q✉❡ ♥❡st❛ ❥♦r♥❛❞❛ ❝♦♠♣❛rt✐❧❤❛r❛♠ s✉❛s ❡①♣❡r✐ê♥❝✐❛s ❡♠ ♣r♦❧ ❞❛ ♥♦ss❛ ❡✈♦❧✉çã♦ ♥♦ ❝✉rs♦✱ ❡♠ ❡s♣❡❝✐❛❧ ❛♦ ♣r♦❢❡ss♦r ❉r ❏♦sé ❈❛r❧♦s✱ ♠❡✉ ♦r✐❡♥t❛❞♦r ♥♦ tr❛❜❛❧❤♦ ❞❡ ❝♦♥❝❧✉sã♦✳

  

❆❣r❛❞❡❝✐♠❡♥t♦s

  ❆♦ Pr♦❢✳ ❉r✳ ❏♦sé ❈❛r❧♦s ❞❡ ▲✐♠❛✱ ♣❡❧❛ s❡r❡♥✐❞❛❞❡ ❡♠ s✉❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧ ❡ ❝♦✲ ♠✉♥✐❝❛çã♦ ❝♦♠ s❡✉s ❛❧✉♥♦s ❡ s✉❛ ✜❧❤❛ ❉♦✉t♦r❛♥❞❛ ❑❛r❧❛ ❦❛t❡r✐♥❡ ❇❛r❜♦③❛ ❞❡ ▲✐♠❛ ♣❡❧♦ ❣r❛♥❞✐♦s♦ ❛♣♦✐♦ ♥❛ ♦r✐❡♥t❛çã♦ ❞♦s ❝♦♠❛♥❞♦s ♥♦ ▲❛t❡①✳

  ❆♦s ♠❡✉s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ ❝❛♠✐♥❤❛❞❛✱ ❆❧❡①❛♥❞r❡ ❋❛r✐❛✱ ❈❧❛r✐❝② ❆❧✈❡s✱ ❈❧❛✉❞✐♦ ❆❣r❛✱ ❉✐♦❣♦ ▼❡✉r❡r✱ ❉❥❛❧♠❛ ●♦♠❡s✱ ❊❞✈❛♥ ❍♦rá❝✐♦✱ ❊✇❡rt♦♥ ❘♦♦s❡✇❡❧t✱ ❋❛❜rí❝✐❛ ❖♠❡♥❛✱ ❋❡r✲ ♥❛♥❞♦ ❱❛❧ér✐♦✱ ●❡r❛❧❞♦ ❇❛rr♦s✱ ●✐❧♠❛r ❚❡♦❞♦③✐♦✱ ●✐✈❛♥✐❧❞♦ ❙❛♥t♦s✱ ●r❛❝✐♥♦ ❋r❛♥❝✐s❝♦✱ ❏♦sé ❆♣❛r❡❝✐❞♦✱ ❏♦sé ❘❡✐♥❛❧❞♦✱ ▲❡❞✐✈❛❧❞♦ ●♦♠❡s✱ ▼❛r❝✐❡❧ ❞❛ ❙✐❧✈❛✱ ❖r❧❛♥❞♦ ❆♣rí❣✐♦✱ ❘♦s❡ ❙✐❧✈❛✱ ❚❤✐❛❣♦ ▲❡ss❛ ❡ ❱❛♥✐♥❛ ❈❛r✈❛❧❤♦ ♣❡❧❛s ❤♦r❛s ❞❡ ❡st✉❞♦s✱ ♣❡❧❛s ✐♥❢♦r♠❛çõ❡s ❡ ❝♦♠✲ ♣r❡❡♥sã♦✱ ❛ss✐♠ ❝♦♠♦✱ àq✉❡❧❡s q✉❡ ✐♥✐❝✐❛r❛♠ ❛ ❥♦r♥❛❞❛ ❝♦♥♦s❝♦✱ ♠❛s ♣♦r ❛❧❣✉♠ ♠♦t✐✈♦✱ ♥ã♦ ❝♦♥❝❧✉ír❛♠✱ ❊❞❝❛r❧♦s ❞❛ ❙✐❧✈❛✱ ❊♠❛♥♦❡❧ ▼❛r✐❛♥♦✱ ●❡♦✈❛♥ ❱✐❡✐r❛✱ ■s❛❛❝ ●♦♠❡s✱ ❏♦sé ❘♦♠✐❧❞♦✱ ❏✉❝é❧✐♦ ❋❡rr❡✐r❛✱ ▲ú❝✐♦ ❆✉❣✉st♦✱ ◆✐❧③❡r ❏❛❝q✉❡❧✐♥❡ ❡ ❚♦♥② ❋á❜✐♦✳

  ➚ ♠✐♥❤❛ ❢❛♠í❧✐❛✱ ❡♠ ❡s♣❡❝✐❛❧✱ ♠✐♥❤❛ ❡s♣♦s❛✱ ✜❧❤♦s✱ s♦❣r❛✱ ❝✉♥❤❛❞❛ ❡ ❝♦✲❝✉♥❤❛❞♦ ♣❡❧♦s ❞✐❛s✱ ♥♦✐t❡s ❡ ✜♥s ❞❡ s❡♠❛♥❛ ❞❡ ❛✉①í❧✐♦✱ q✉❡ ❢♦✐ ❡ss❡♥❝✐❛❧ ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ss❡ ❝✉rs♦✳

  ❆♦s ❝♦♠♣❛♥❤❡✐r♦s ❞❡ tr❛❜❛❧❤♦ q✉❡ ♠❡ ❞❡r❛♠ ❛♣♦✐♦ ❝♦♠ ❛ ♣❛rt❡ ♣rát✐❝❛ ❞❡ss❡ tr❛✲ ❜❛❧❤♦✱ ❆❧♠✐r✱ ❇r✉♥✐♦ ❆❧❜✐♥♦✱ ❈❛✐♦ ❈és❛r✱ ▲✉❝✐✈❛❧❞❛ ❇❛r❜♦③❛✱ ❘✐❧❞s♦♥✱ ❙ér❣✐♦ ❈❛❜r❛❧ ❡ ❲✐t❡♠❜❡r❣✱ ❛ss✐♠ ❝♦♠♦ ✉♠ ❛❧✉♥♦ ❡♠ ❡s♣❡❝✐❛❧ ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ❉r ❋❡r♥❛♥❞❡s ▲✐♠❛✱ ❆❧✐ss♦♥ ▲♦♣❡s✱ q✉❡ ♣♦ss✉✐ ✉♠ ❢✉t✉r♦ ♣r♦♠✐ss♦r ♥❛ ár❡❛ ❞❛s ❡①❛t❛s✳

  ❆♦s ♠❡✉s ❛❧✉♥♦s ❞❛s ❊s❝♦❧❛s ❊st❛❞✉❛❧ ❉r ❋❡r♥❛♥❞❡s ▲✐♠❛✱ ▼✉♥✐❝✐♣❛❧ ◆✐s❡ ❞❛ ❙✐❧✈❡✐r❛ ❡ ❞♦ ❈♦❧é❣✐♦ ❆♥❝❤✐❡t❛✱ ♣♦r ❝♦♥tr✐❜✉✐r❡♠✱ ❞❡ ❛❧❣✉♠❛ ❢♦r♠❛✱ ❝♦♠ ❡ss❛ ♠✐♥❤❛ ❝♦♥q✉✐st❛✳

  

❘❡s✉♠♦

  ▼✉✐t♦s sã♦ ♦s ❞❡s❛✜♦s q✉❡ ❛ ▼❛t❡♠át✐❝❛ ♣r♦♣õ❡ ❞✉r❛♥t❡ t♦❞❛ ❛ ♥♦ss❛ ✈✐❞❛ ♣r♦✜ss✐♦♥❛❧✱ s❡❥❛ ❝♦♠♦ ❡st✉❞❛♥t❡ ♦✉ ❝♦♠♦ ♣r♦✜ss✐♦♥❛❧ ❞❛ ár❡❛✳ ❆ ❜✉s❝❛ ♣❡❧♦ ❝♦♥❤❡❝✐♠❡♥t♦ é ♥❡❝❡ssár✐❛ ♣❛r❛ ♣♦❞❡r ❞❛r♠♦s ❝♦♥t❛ ❞❛s ♥♦✈✐❞❛❞❡s q✉❡ s✉r❣❡♠ ❛ ❝❛❞❛ ❞✐❛✳ ❙❡♠♣r❡ ♣r❡❝✐s❛♠♦s ❞❡ ♠♦t✐✈❛çã♦ ♣❛r❛ ♣♦❞❡r ✜❝❛r♠♦s ❛t✉❛❧✐③❛❞♦s ❝♦♠ ♦s ❛ss✉♥t♦s ♥❡❝❡ssár✐♦s ♣❛r❛ ♦ ❡♥t❡♥❞✐♠❡♥t♦ ❞❡ss❛s ♥♦✈✐❞❛❞❡s✳ ❆ s❡❣✉✐r✱ ♠♦str❛r❡♠♦s ✉♠ ♠❛t❡r✐❛❧ q✉❡ ♠♦t✐✈❛rá ♦ ♣r♦❢❡ss♦r ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ ❜✉s❝❛ ♦ ❛♣r❡♥❞✐③❛❞♦ ♦ t❡♠♣♦ t♦❞♦✱ r❡s❣❛t❛♥❞♦ ♦ ❡stí♠✉❧♦ q✉❡ ♠✉✐t❛s ✈❡③❡s s❡ ♣❡r❞❡ ♣❡❧♦ ❢❛t♦ ❞♦ ♣r♦✜ss✐♦♥❛❧ ❞❛ ❡❞✉❝❛çã♦ ♥ã♦ ❝♦♥s❡❣✉✐r r❡s♦❧✈❡r ❛❧❣✉♥s ♣r♦❜❧❡♠❛s ♠❛t❡♠át✐❝♦s q✉❡ s✉r❣❡♠ ♥❛ s✉❛ ❝❛♠✐♥❤❛❞❛✳ ❖s ❛ss✉♥t♦s q✉❡ s❡rã♦ ❛❜♦r❞❛❞♦s ❛q✉✐ sã♦ ❞❡ ❣r❛♥❞❡ ✉t✐❧✐❞❛❞❡ ♥❛ ❊❞✉❝❛çã♦ ❇ás✐❝❛ ❡ ♣♦r ❡❧❡s t❡r❡♠ ✉♠❛ ❜❡❧❡③❛ ❡ ❣r❛❝✐♦s✐❞❛❞❡ ♥❛ r❡s♦❧✉çã♦ ❞♦s ♣r♦❜❧❡♠❛s ❞❡✈❡♠ s❡r ❛♣r❡❝✐❛❞♦s ❝♦♠ ♠❛✐♦r ✐♥t❡r❡ss❡✳ ❆ ❧✐♥❣✉❛❣❡♠ ✉t✐❧✐③❛❞❛ ❛q✉✐ é ♠✉✐t♦ ❛❣r❛❞á✈❡❧ ♣❛r❛ ♦ ♣r♦❢❡ss♦r ❞❡ ▼❛t❡♠át✐❝❛✱ ❡st✐♠✉❧❛♥❞♦✲♦ ❛✐♥❞❛ ♠❛✐s ❡♠ q✉❡r❡r ❛❞q✉✐r✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✳ ❊st❡ ♠❛t❡r✐❛❧ ❞❡✈❡ s❡r ✉s❛❞♦ ❞❡ ♠❛♥❡✐r❛ ✐♥t❡♥s❛ ❡ ♦st❡♥s✐✈❛ ♣❛r❛ q✉❡ ✈♦❝ê ❝♦♠♦ ♣r♦❢❡ss♦r t❡♥❤❛ t♦t❛❧ ❝♦♥✜❛♥ç❛ ❡ ❞♦♠í♥✐♦ ❞♦ ❝♦♥t❡ú❞♦ ♣❛r❛ ♣r♦♣♦r té❝♥✐❝❛s ❡ ❡str❛té❣✐❛s ♥♦s ♣❧❛♥❡❥❛♠❡♥t♦s ❞❡ s✉❛s ❛✉❧❛s✳

  

❆❜str❛❝t

  ❚❤❡r❡ ❛r❡ ♠❛♥② ❝❤❛❧❧❡♥❣❡s t❤❛t ♠❛t❤❡♠❛t✐❝s ♦✛❡rs t❤r♦✉❣❤♦✉t ♦✉r ✇♦r❦✐♥❣ ❧✐✈❡s✱ ❡✐t❤❡r ❛s ❛ st✉❞❡♥t ♦r ❛s ❛♥ ❛r❡❛ ♦❢ ✇♦r❦✳ ❚❤❡ q✉❡st ❢♦r ❦♥♦✇❧❡❞❣❡ ✐s ♥❡❝❡ss❛r② t♦ ❜❡ ❛❜❧❡ t♦ ❣✐✈❡ ❛❝❝♦✉♥t ♦❢ t❤❡ ♥❡✇ ❞❡✈❡❧♦♣♠❡♥ts t❤❛t ❛r✐s❡ ❡✈❡r② ❞❛②✳ ❆❧✇❛②s ♥❡❡❞ ♦❢ ♠♦t✐✈❛t✐♦♥ ✐♥ ♦r❞❡r t♦ st❛② ✉♣❞❛t❡❞ ✇✐t❤ t❤❡ ♠❛tt❡rs ♥❡❝❡ss❛r② ❢♦r t❤❡ ✉♥❞❡rst❛♥❞✐♥❣ ♦❢ t❤❡s❡ ♥❡✇s✳ ◆❡①t✱ ✇❡ ✇✐❧❧ s❤♦✇ ❛ ♠❛t❡r✐❛❧ t❤❛t ✇✐❧❧ ♠♦t✐✈❛t❡ t❤❡ ♠❛t❤ t❡❛❝❤❡r ✇❤♦ s❡❡❦s ❧❡❛r♥✐♥❣ ❛❧❧ t❤❡ t✐♠❡✱ r❡s❝✉✐♥❣ t❤❡ st✐♠✉❧✉s t❤❛t ♦❢t❡♥ ❣❡ts ❧♦st ❜❡❝❛✉s❡ ♦❢ ♣r♦❢❡ss✐♦♥❛❧ ❡❞✉❝❛t✐♦♥ ❝❛♥ ♥♦t s♦❧✈❡ s♦♠❡ ♠❛t❤❡♠❛t✐❝❛❧ ♣r♦❜❧❡♠s t❤❛t ❛r✐s❡ ✐♥ ②♦✉r ✇❛❧❦✳ ❚❤❡ s✉❜❥❡❝ts t♦ ❜❡ ❞✐s❝✉ss❡❞ ❤❡r❡ ❛r❡ ♦❢ ❣r❡❛t ✉s❡ ✐♥ ❜❛s✐❝ ❡❞✉❝❛t✐♦♥ ❛♥❞ t❤❡② ❤❛✈❡ ❛ ❜❡❛✉t② ❛♥❞ ❣r❛❝❡ ✐♥ s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ♠✉st ❜❡ ❡①❛♠✐♥❡❞ ✇✐t❤ ❣r❡❛t ✐♥t❡r❡st✳ ❚❤❡ ❧❛♥❣✉❛❣❡ ✉s❡❞ ❤❡r❡ ✐s ✈❡r② ♣❧❡❛s❛♥t ❢♦r t❤❡ t❡❛❝❤❡r ♦❢ ▼❛t❤❡♠❛t✐❝s✱ st✐♠✉❧❛t✐♥❣ ✐t ❡✈❡♥ ♠♦r❡ ✐♥ ✇❛♥t✐♥❣ t♦ ❛❝q✉✐r❡ ❦♥♦✇❧❡❞❣❡✳ ❚❤✐s ♠❛t❡r✐❛❧ s❤♦✉❧❞ ❜❡ ✉s❡❞ ✐♥ ❛♥ ✐♥t❡♥s❡ ❛♥❞ ♦✈❡rt ✇❛② ❢♦r ②♦✉ ❛s ❛ t❡❛❝❤❡r ❤❛✈❡ ❢✉❧❧ ❝♦♥✜❞❡♥❝❡ ❛♥❞ ❝♦♥t❡♥ts ♦❢ t❤❡ ❞♦♠❛✐♥ t♦ ♣r♦♣♦s❡ t❡❝❤♥✐q✉❡s ❛♥❞ str❛t❡❣✐❡s ✐♥ ♣❧❛♥♥✐♥❣ t❤❡✐r ❧❡ss♦♥s✳

  

❙✉♠ár✐♦

  ✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✹

  ✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺ ✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸

  ✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ❞♦s ◆ú♠❡r♦s ■♥t❡✐r♦s ✶✽ ✷✳✶ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✷ ❉✐✈✐sã♦ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✸ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ❡ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵

  ✷✳✸✳✶ ▼á①✐♠♦ ❉✐✈✐s♦r ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✷ Pr♦♣r✐❡❞❛❞❡s ❞♦ ♠❞❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✸✳✸ ▼í♥✐♠♦ ▼ú❧t✐♣❧♦ ❈♦♠✉♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹

  ✷✳✹ ◆ú♠❡r♦s Pr✐♠♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✺ ❆r✐t♠ét✐❝❛ ❞♦s ❘❡st♦s ❡ ❙✉❛s ❆♣❧✐❝❛çõ❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵

  ✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✸✾ ✸✳✶ ❈♦♥❣r✉ê♥❝✐❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✾ ✸✳✷ ❆❧❣✉♠❛s ❆♣❧✐❝❛çõ❡s ❞❡ ❈♦♥❣r✉ê♥❝✐❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✹ ✸✳✸ ❆ ❉❡s✐❣✉❛❧❞❛❞❡ ❚r✐❛♥❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✽ ✸✳✹ ◗✉❛❞r✐❧át❡r♦s ◆♦tá✈❡✐s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✶

  ✸✳✹✳✶ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✷ ✸✳✹✳✷ ❚r❛♣é③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✺ ✸✳✹✳✸ ❘❡tâ♥❣✉❧♦✱ ❧♦s❛♥❣♦ ❡ q✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✺✼

  ✸✳✺ ▲✉❣❛r❡s ●❡♦♠étr✐❝♦s ❇ás✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✵ ✸✳✻ ❚❡♦r❡♠❛ ❞❡ ❚❤❛❧❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✷ ✸✳✼ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✻✻ ✸✳✽ ❆❧❣✉♥s ❚❡♦r❡♠❛s ❞❡ ❙❡♠❡❧❤❛♥ç❛ ❞❡ ❚r✐â♥❣✉❧♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✼✷ ✸✳✾ ➪r❡❛s ❞❡ ❋✐❣✉r❛s P❧❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✷

  ❙❯▼➪❘■❖ ✈✐✐✐

  ✸✳✾✳✶ ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✾✳✷ ➪r❡❛ ❞♦ ❚r✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✹ ✸✳✾✳✸ ➪r❡❛ ❞♦ ❚r❛♣é③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✺ ✸✳✾✳✹ Pr♦♣r✐❡❞❛❞❡s ●❡♦♠étr✐❝❛s ■♠♣♦rt❛♥t❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✻

  ✸✳✶✵ ➪r❡❛ ❞♦ ❈ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽✾ ✸✳✶✵✳✶ ➪r❡❛s ❞❡ s❡t♦r❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✵

  ✸✳✶✶ Pr♦❜❧❡♠❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾✷ ✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠

  ✾✾ ✺ ❙♦❧✉çõ❡s ❞♦s Pr♦❜❧❡♠❛s

  ✶✶✵ ✺✳✶ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡♠❡♥t❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵

  ✺✳✶✳✶ Pr✐♥❝í♣✐♦ ❞❛ ■♥❞✉çã♦ ❋✐♥✐t❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶✵ ✺✳✶✳✷ Pr✐♥❝í♣✐♦ ❞❛ ❈❛s❛ ❞♦s P♦♠❜♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✵

  ✺✳✷ ❆❧❣✉♥s ❆s♣❡❝t♦s ❞❛ ❉✐✈✐s✐❜✐❧✐❞❛❞❡ ♥❛ ➪❧❣❡❜r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷✹ ✺✳✸ ❆❧❣✉♥s ❚ó♣✐❝♦s ❞❡ ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸✽ ✺✳✹ Pr✐♥❝í♣✐♦ ❋✉♥❞❛♠❡♥t❛❧ ❞❡ ❈♦♥t❛❣❡♠ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺✺

  ✻ ❈♦♥s✐❞❡r❛çõ❡s ✜♥❛✐s ✶✻✹

  ❇✐❜❧✐♦❣r❛✜❛ ✶✻✼

  

■♥tr♦❞✉çã♦

  ❊ss❡ tr❛❜❛❧❤♦ ✈✐s❛ ❝♦♥t❡♠♣❧❛r ♣r♦❢❡ss♦r❡s ❞❡ ▼❛t❡♠át✐❝❛ ❡ ár❡❛s ❛✜♥s✱ ❛tr❛✈és ❞❡ ❝✉rs♦s ❞❡ ❢♦r♠❛çã♦ ♣❛r❛ ♣r♦❢❡ss♦r❡s✱ ♠♦t✐✈❛♥❞♦✲♦s ♣❛r❛ ✉♠ ♠❡❧❤♦r ❞❡s❡♠♣❡♥❤♦ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳ ❊st❡ tr❛❜❛❧❤♦ ❢❛③ ♣❛rt❡ ❞❡ ✉♠ s❡♥t✐♠❡♥t♦ ❞❡ ❛♥❣úst✐❛ q✉❡ t✐✈❡ ❤á ❛❧❣✉♥s ❛♥♦s✱ q✉❛♥❞♦ ♣❡r❝❡❜✐ q✉❡ ❡st❛✈❛ ❡♥s✐♥❛♥❞♦ ▼❛t❡♠át✐❝❛ ❞❡ ♠♦❞♦ ❡q✉✐✈♦❝❛❞♦✱ ❝♦♠ r❡s♦❧✉çõ❡s ❞❡ ❡①❡r❝í❝✐♦s✱ s❡♠ ♠✉✐t❛ ♣r❡♦❝✉♣❛çã♦ ❝♦♠ ❞❡✜♥✐çõ❡s✱ ♣r♦♣r✐❡❞❛❞❡s✱ t❡♦r❡♠❛s ❡ ❛♣❧✐❝❛çõ❡s ❞❡✈✐❞❛s ❞❡ss❡s ❝♦♥❤❡❝✐♠❡♥t♦s✳

  ❊ss❡ s❡♥t✐♠❡♥t♦ ♠❡ ❡st✐♠✉❧♦✉ ❛ ❞❡s❡♥✈♦❧✈❡r ❡ss❡ tr❛❜❛❧❤♦✱ ♣♦✐s ♣❡♥s♦ q✉❡ ❡ss❛ ❛♥❣úst✐❛ ♥ã♦ é ♣r✐✈✐❧é❣✐♦ ♠❡✉✳ P♦rt❛♥t♦✱ ♠❡✉s ❝♦❧❡❣❛s t❛♠❜é♠ ❞❡✈❡♠ ♣❛ss❛r ♣♦r ✐ss♦✳ ❊ ❛♥t❡s ❞✐ss♦ s❡ ♣r♦❧♦♥❣❛r ♣❛r❛ ❡❧❡s✱ ❣♦st❛r✐❛ ❞❡ ❝♦♠♣❛rt✐❧❤❛r ❛s ✐❞❡✐❛s ❞❡ss❡ tr❛❜❛❧❤♦✱ q✉❡ ❡s♣❡r♦ s❡r ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ ♣❛r❛ ♠❡✉s ❝♦♠♣❛♥❤❡✐r♦s✳ ❞❡ ❡♥❝♦r❛❥á✲❧♦ ❛♦s ❞❡s❛✜♦s q✉❡ ❛ ▼❛t❡♠át✐❝❛ ♣r♦♣õ❡ ♣❛r❛ ❛ ❣❡♥t❡✳ P♦r ♠✉✐t❛s ✈❡③❡s ♣❛ss❛♠♦s ♣♦r s✐t✉❛çõ❡s q✉❡ ♥♦s ❞❡✐①❛♠ ❝♦♥str❛♥❣✐❞♦✱ ♣❡❧♦ ❢❛t♦ ❞❡ q✉❡ ♥❛q✉❡❧❡ ♠♦♠❡♥t♦ ❞❡ ❞✐✜❝✉❧❞❛❞❡ ♥ã♦ ❝♦♥s✐❣❛♠♦s ♦❜t❡r ✉♠❛ ✐❞❡✐❛ ♣❛r❛ ❛ r❡s♦❧✉çã♦ ❞❡ ✉♠ ♣r♦❜❧❡♠❛ ♣r♦♣♦st♦✳

  P❛r❛ t♦❞♦ ❝♦♠❡♥tár✐♦ q✉❡ s❡ ♦✉✈❡✱ ❞❡✈❡♠♦s t❡r ❛t❡♥çã♦ ❡♠ ❝❛❞❛ ♣❛❧❛✈r❛ q✉❡ s❡ é ❢❛❧❛❞❛✳ ▼✉✐t❛s ❞❡ss❛s ♣❛❧❛✈r❛s ♣♦❞❡♠ ♥♦s ❧❡✈❛r ❛ ❞✐✈❡rs♦s ❝❛♠✐♥❤♦s✱ ❜❛st❛ ✈♦❝ê ❛ss✐♠✐❧❛r ❛s ✐♥❢♦r♠❛çõ❡s q✉❡ sã♦ út❡✐s ♣❛r❛ ❞❡s❡♥✈♦❧✈❡r ❞❡t❡r♠✐♥❛❞♦s ❛ss✉♥t♦s✳ ❱♦✉ ❝♦♥t❛r ✉♠ ❡①❡♠♣❧♦ q✉❡ ❛❝♦♥t❡❝❡✉ ❝♦♠✐❣♦✳ ❊✉ ❡st❛✈❛ ❛ss✐st✐♥❞♦ ✉♠❛ ❛✉❧❛ ❞❡ ❣❡♦♠❡tr✐❛✱ q✉❛♥❞♦ ♦ ♣r♦❢❡ss♦r ❝♦❧♦❝♦✉ ♥♦ q✉❛❞r♦ ✉♠ tr✐â♥❣✉❧♦ ❝♦♠ ♦s â♥❣✉❧♦s ❞❡ ♠❡❞✐❞❛s x✱ 2x ❡ 3x ♣❛r❛ ❡①♣❧✐❝❛r ❛❧❣✉♠ t❡♦r❡♠❛ ♦✉ ❛❧❣✉♠❛ ♣r♦♣r✐❡❞❛❞❡✱ q✉❛♥❞♦ ✉♠ ❞♦s ♠❡✉s ❝♦❧❡❣❛s ❝❤❛♠♦✉ ❛ ♠✐♥❤❛ ❛t❡♥çã♦ ❛♦ ❞✐③❡r ✲ ❊ss❡ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦ ✲ ♠❛s ♥ã♦ ❡①♣❧✐❝♦✉ ♦ ♣♦rq✉❡✳ ❋✐q✉❡✐✱ ♣♦r q✉❛s❡ ✷✵ ♠✐♥✉t♦s✱ s❡♠ ❡♥t❡♥❞❡r ❛q✉❡❧❛ ✐♥❢♦r♠❛çã♦ q✉❡ ❡❧❡ ♣❛ss♦✉✱ ❞❡✐①❛♥❞♦✲ ♠❡ ♣❡♥s❛t✐✈♦ ❞❡ ❝♦♠♦ é q✉❡ ❡❧❡ ❡♥①❡r❣❛ tã♦ r❛♣✐❞❛♠❡♥t❡ q✉❡ ❛q✉❡❧❡ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳ ◆❛ ✈❡r❞❛❞❡✱ ♣♦❞❡♠♦s ♣r♦✈❛r ✐ss♦ ❞❡ ❞✐✈❡rs❛s ♠❛♥❡✐r❛s✱ ♠❛s ❛s ❞❡✜♥✐çõ❡s q✉❡ ♥ã♦ sã♦ ❝♦❧♦❝❛❞❛s ❝♦♠ ♠✉✐t❛ ❛✜♥✐❞❛❞❡ ❡♠ ♥♦ss❛ ♠❡♥t❡✱ ♥♦s ❞❡✐①❛♠ ❝❡❣♦ q✉❛♥t♦ ❛ ❡ss❛ ✐♥❢♦r♠❛çã♦✳

  ✆

  ➱ ❢á❝✐❧ ✈❡r q✉❡ s❡♥❞♦ ❛ s♦♠❛ ❞♦s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r s❡♠♣r❡ é 180 ✱

  ❙❯▼➪❘■❖ ✷

  ✆ ✆ ✆

  ❡♥tã♦ x 2x 3x ✏ 180 ✱ ♦ q✉❡ ✐♠♣❧✐❝❛ q✉❡ x ✏ 30 ✱ ❡ ❞❡ss❛ ❢♦r♠❛ 3x ✏ 90 ❡ ♣♦rt❛♥t♦✱ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳ P❡♥s❡✐ ✐ss♦ ❛ ♣r✐♥❝í♣✐♦✱ ♠❛s ❛❝❤♦ q✉❡ ♣❡❧❛ ❛❣✐❧✐❞❛❞❡ q✉❡ ❡❧❡ ❞❡✉ ❛ r❡s♣♦st❛✱ ❛❝r❡❞✐t♦ q✉❡ ♥ã♦ ❢♦✐ ❡ss❛ ❝♦♥t❛ q✉❡ ❡❧❡ ❢❡③ ♥❛ ❤♦r❛✳ ▲♦❣♦ ❡♠ s❡❣✉✐❞❛✱ ❡ss❡ ♠❡s♠♦ ❝♦♠♣❛♥❤❡✐r♦ ❢❡③ ❛ s❡❣✉✐♥t❡ ❛✜r♠❛çã♦✿ ❙❡ ❛ s♦♠❛ ❞❡ ❞♦✐s â♥❣✉❧♦s ✐♥t❡r♥♦s ❞❡ ✉♠ tr✐â♥❣✉❧♦ q✉❛❧q✉❡r ❢♦r ✐❣✉❛❧ ❛♦ t❡r❝❡✐r♦✱ ❡♥tã♦ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳ ❊ ❡ss❛ ❝♦♥t❛ é ❜❡♠ ♠❛✐s ❢á❝✐❧ ❞❡ ❢❛③❡r ❞♦ q✉❡ ❛ ♣r✐♠❡✐r❛✱ ♣♦✐s x 2x ✏ 3x✱ ♣♦rt❛♥t♦ ♦ tr✐â♥❣✉❧♦ é r❡tâ♥❣✉❧♦✳

  ◆❛ é♣♦❝❛ ❞♦s ●r❡❣♦s✱ ♣♦❞✐❛✲s❡ ❢❛❧❛r ❞♦ ❝á❧❝✉❧♦ ❡ ❞❛ ❣❡♦♠❡tr✐❛ ❝♦♠♦ ♣❛rt❡s ú♥✐❝❛s ❞❡ ✉♠ ❝♦r♣♦ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s ❜❡♠ ❞❡❧✐♠✐t❛❞♦ ❡ ♥ã♦ ♠✉✐t♦ ❡①t❡♥s♦✳ ❍♦❥❡ ❡♠ ❞✐❛✱ ♣♦ré♠✱ ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝♦♥t❡ú❞♦s ❞❡ ♠❛t❡♠át✐❝❛ q✉❡ s❡ ❝♦♥❤❡❝❡ é ✐♠❡♥s❛ ❡ ❝r❡s❝❡ ❡①♣♦♥❡♥❝✐✲ ❛❧♠❡♥t❡✱ t♦r♥❛♥❞♦✲s❡ ❞✐❢í❝✐❧ ❞❡❝✐❞✐r q✉❛❧ ❞❡✈❡ s❡r ❛ ♠❛t❡♠át✐❝❛ q✉❡ s❡ ❛❝♦♥s❡❧❤❡ ❡♥s✐♥❛r ❡ ❝♦♠♦ ❞❡✈❡ s❡r ❛♣r❡s❡♥t❛❞❛ ♣❛r❛ s✉❛ ♠❡❧❤♦r ❝♦♠♣r❡❡♥sã♦ ❡ s✉❛ ♠❡❧❤♦r ✉t✐❧✐❞❛❞❡ ♣❛r❛ ♦ ❢✉t✉r♦ ❞♦s ❛❧✉♥♦s ❬✾❪✳ ❊✉ ❛❝r❡s❝❡♥t♦✱ ❛✐♥❞❛✱ ❛ ❡ss❛ ❡✈♦❧✉çã♦ q✉❡ ❞✐❛ ❛♣ós ❞✐❛ ❛♣❛r❡✲ ❝❡♠ ♥♦✈❛s t❡❝♥♦❧♦❣✐❛s s❡♥❞♦ q✉❡❜r❛❞❛s ♣r♦♣r✐❡❞❛❞❡s q✉❡ ❛♥t❡s ❡r❛♠ ✐♥q✉❡❜rá✈❡✐s ❡ q✉❡ ♥ós✱ ❞♦❝❡♥t❡s✱ ❛❝❤❛♠♦s q✉❡ s❛❜❡♠♦s ✉♠ ♣♦✉❝♦ ❞❡ ♠❛t❡♠át✐❝❛✱ ♠❛s q✉❡ ♥♦ ❢✉♥❞♦ s♦♠♦s ✐♥s❡❣✉r♦s ❞❛q✉✐❧♦ q✉❡ tr❛♥s♠✐t✐♠♦s ❡♠ s❛❧❛ ❞❡ ❛✉❧❛ ❡ ♣r♦♣♦♠♦s ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ ♥ã♦ sã♦ ❞❡ ❣r❛♥❞❡ ✈❛❧✐❛ ♣❛r❛ ♦ ❛❧✉♥♦✱ t♦r♥❛♥❞♦✲♦ ✉♠ ❢r✉str❛❞♦ ❝♦♠ ❛ ❞✐s❝✐♣❧✐♥❛✳ ❖ ♣r♦❢❡ss♦r ❲❛❧t❡r ❙♣✐♥❡❧❧✐✱ ❡♠ ❡♥tr❡✈✐st❛ ❛ ❘❡✈✐st❛ ❈á❧❝✉❧♦ ❞❡❝❧❛r❛ q✉❡ s❡♥t❡ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡ ♥♦s t❡♠♣♦s ❞❡ ❤♦❥❡ ❞♦ q✉❡ ❤á ❞♦✐s ❛♥♦s ❡ ❛✐♥❞❛ ♠❛✐s ❞✐✜❝✉❧❞❛❞❡ ❞♦ q✉❡ ❤á q✉❛tr♦ ❛♥♦s✳

  ▲❡♠❜r♦✲♠❡ q✉❡ ❛♦ ❡st✉❞❛r ❣❡♦♠❡tr✐❛ ❛♥❛❧ít✐❝❛ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♠❛✐s ❡s♣❡❝✐✜❝❛♠❡♥t❡ ♦ ❡st✉❞♦ ❞❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛✱ ♣❡❞✐ ♣❛r❛ ♦ ♣r♦❢❡ss♦r q✉❡ ❞❡♠♦♥str❛ss❡ ❛ ❡q✉❛çã♦ ❞❛ ❝✐r❝✉♥❢❡rê♥❝✐❛ ❡①♣♦st❛ ♣♦r ❡❧❡ ♥♦ q✉❛❞r♦ ♥❛q✉❡❧❛ é♣♦❝❛✳ ❆ r❡s♣♦st❛ q✉❡ ❡✉ ♦❜t✐✈❡ ❞❡ss❡ ♣r♦❢❡ss♦r✱ q✉❡ ❡r❛ ❝♦♥s✐❞❡r❛❞♦ ♦ ♠❡❧❤♦r ❞♦ ❝♦❧é❣✐♦ ❡ ✉♠ ❞♦s ♠❡❧❤♦r❡s ❞♦ ❡st❛❞♦✱ ❢♦✐ q✉❡ ❛ ❞❡♠♦♥str❛çã♦ ❞❡ss❛ ❡q✉❛çã♦ ❞❡♠♦r❛r✐❛ ♥♦ ♠í♥✐♠♦ ❞✉❛s ❛✉❧❛s ❞❡ ✺✵ ♠✐♥✉t♦s ❡ q✉❡ ♥ã♦ ✈❛❧❡r✐❛ ❛ ♣❡♥❛ ❞❡♠♦♥strá✲❧❛✳ ❋✐q✉❡✐ ❝♦♠ ✐ss♦ ❣✉❛r❞❛❞♦ ❡♠ ♠✐♥❤❛ ❝❛❜❡ç❛✱ ❛❝❤❛♥❞♦ q✉❡ ✉♠ ❞✐❛ ♣♦❞❡r✐❛ ❞❡❞✉③í✲❧❛✱ ♣♦✐s ❥á ❤❛✈✐❛ ❢❡✐t♦ ✐ss♦ ❝♦♠ ❛ ❞✐stâ♥❝✐❛ ❡♥tr❡ ❞♦✐s ♣♦♥t♦s✳ ❍♦❥❡✱ ❡✉ ♣❡r❝❡❜♦ q✉❡ ♥ã♦ s❡r✐❛ ♥❡❝❡ssár✐❛ ❞✉❛s ❛✉❧❛s ♣❛r❛ ✉♠❛ ❞❡♠♦♥str❛çã♦ ❞❡ ✉♠❛ ❡q✉❛çã♦ ❞❡ss❛✱ ♣♦✐s ✉s❛ ❛♣❡♥❛s ✉♠ ♣♦✉❝♦ ❞❡ ❣❡♦♠❡tr✐❛ ❡ ♦ ❚❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s✳ ❖✉ s❡❥❛✱ ✉♠❛ ❝♦✐s❛ s✐♠♣❧❡s ❡ ❛tr❛t✐✈❛ ♣❛r❛ ♠✐♠✳ ❚❡♥❤♦ ❝❡rt❡③❛ ❞❡ q✉❡ s❡ ❡ss❡ ♣r♦❢❡ss♦r ❝♦❧♦❝❛ss❡ ✐ss♦ ❝♦♠♦ ✉♠ ❞❡s❛✜♦ ♣❛r❛ ♠✐♠✱ ❡✉ ♣♦❞❡r✐❛ ♠❡ ❛♣r♦①✐♠❛r ♠❡❧❤♦r ❞❛ ▼❛t❡♠át✐❝❛ ❡ ♠❡ ❛♣❛✐①♦♥❛r ❛✐♥❞❛ ♠❛✐s ♣♦r ❡❧❛✳ P♦ré♠ ❝❛❞❛ ❝♦✐s❛ ❡♠ s❡✉ t❡♠♣♦✱ ❡ ❡✉ ❛❝❤♦ q✉❡ ♦ ♠❡✉ t❡♠♣♦✱ ❛♦s ♣♦✉❝♦s ❡stá ❝❤❡❣❛♥❞♦✳

  ❖ ❡♥s✐♥♦ ❞❡ ▼❛t❡♠át✐❝❛ ❤♦❥❡ ♥♦ ❇r❛s✐❧ ❞✐❢❡r❡ ♣♦✉❝♦ ❞♦ ❡♥s✐♥♦ ♣r❛t✐❝❛❞♦ ❤á ✷✵ ❛♥♦s✳ ❆ ❝❛❞❛ ❛♥♦✱ ❧✐✈r♦s ♥♦✈♦s sã♦ ❡❞✐t❛❞♦s r❡♣❡t✐♥❞♦ q✉❛s❡ s❡♠♣r❡ ♦ ♠❡s♠♦ ❡st✐❧♦ ❡ ♦s ♠❡s♠♦s ❝♦♥t❡ú❞♦s ❞♦s ❛♥t❡r✐♦r❡s✳ ❊①✐st❡♠ ❤♦❥❡ ♥♦ ❇r❛s✐❧ ❜♦♥s ❧✐✈r♦s ❞❡ ▼❛t❡♠át✐❝❛ ❞❡❞✐❝❛❞♦s ❛♦s ❛❧✉♥♦s t❛♥t♦ ❞♦ ❡♥s✐♥♦ ❢✉♥❞❛♠❡♥t❛❧ q✉❛♥t♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❊♥tr❡t❛♥t♦✱ ♦ q✉❡ ❧❤❡s ❢❛❧t❛ é ✉♠ ✐♥❣r❡❞✐❡♥t❡ q✉❡✱ ♥♦ ♠✉♥❞♦ ❞❡ ❤♦❥❡✱ é ❢✉♥❞❛♠❡♥t❛❧✿ ♦ ❡stí♠✉❧♦ à ❝r✐❛t✐✈✐❞❛❞❡✳

  ❙❯▼➪❘■❖ ✸

  ❊♥t❡♥❞❡♠♦s q✉❡ ♥ã♦ é s✉✜❝✐❡♥t❡ ♣❛r❛ ❛ ❢♦r♠❛çã♦ ❞♦ ❢✉t✉r♦ ❝✐❞❛❞ã♦ ✉♠ ❛♣r❡♥❞✐③❛❞♦ ❜✉r♦❝rát✐❝♦ ❞❛ ▼❛t❡♠át✐❝❛ ❡ ♣❡r❝❡❜❡♠♦s ❛ ✐♠♣♦rtâ♥❝✐❛ ❞❡ ❡st✐♠✉❧❛r ♦s ❛❧✉♥♦s ❞❡s❞❡ ❝❡❞♦ ❛ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s ♥♦✈♦s ❡ ❞❡s❛✜❛♥t❡s✱ ♣r♦♣✐❝✐❛♥❞♦ ♦ ❞❡s❡♥✈♦❧✈✐♠❡♥t♦ ❞❛ ✐♠❛❣✐♥❛çã♦ ❡ ❞❛ ❝r✐❛t✐✈✐❞❛❞❡ ❬✶✵❪✳ ❖ ❝♦♥❤❡❝✐♠❡♥t♦ é ❛❞q✉✐r✐❞♦ ❛tr❛✈és ❞❛ ✈♦♥t❛❞❡ ❞❡ ❛♣r❡♥❞❡r ❞♦ ❛❧✉♥♦✳ ◆ã♦ ❡①✐st❡ ✉♠ ♣r♦❢❡ss♦r q✉❡ ❝♦♥s✐❣❛ tr❛♥s♠✐t✐r ❝♦♥❤❡❝✐♠❡♥t♦ ♣❛r❛ ♦ ❛❧✉♥♦ s❡ ❡ss❡ ♥ã♦ ❛❞♠✐t❡ ♦ ❛♣r❡♥❞✐③❛❞♦✳ ❉❡ss❛ ❢♦r♠❛ é ♠✉✐t♦ ❝ô♠♦❞♦ ♣❛r❛ q✉❡♠ ❡st❛r ❛♣r❡♥❞❡♥❞♦✱ ♣♦✐s ♥ã♦ t❡r✐❛ ❡s❢♦rç♦ ♥❡♥❤✉♠ ♣❛r❛ ❛❞q✉✐r✐r ♦ ❝♦♥❤❡❝✐♠❡♥t♦✱ ❡ ❛ss✐♠✱ ♥ã♦ ❞❛r✐❛ ✈❛❧♦r ♣❛r❛ ❛q✉✐❧♦ q✉❡ ❡st❛✈❛ s❡♥❞♦ ❛ss✐♠✐❧❛❞♦✳

  ❆❧é♠ ❞✐ss♦✱ ♣❡♥s♦ q✉❡ ♦ ❛❧✉♥♦ ❞❡✈❡ ❢❛③❡r ♦ s❡✉ ♣❛♣❡❧ ❞❡♥tr♦ ❞❡ s✉❛ ♣r♦✜ssã♦ ❞❡ ❡st✉❞❛♥t❡✳ ❖ q✉❡ q✉❡r♦ ❞✐③❡r é q✉❡ s❡ ♦ ❛❧✉♥♦ ♥ã♦ ❢♦r ❛trás ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ♣♦r ✈♦♥t❛❞❡ ♣ró♣r✐❛✱ ♥ã♦ ❡①✐st✐rá ✉♠❛ ♣❡ss♦❛ q✉❡ ❝♦♥s✐❣❛ ❝♦❧♦❝❛r ✐ss♦ ♥❡❧❡✳ ❖ ♣❛♣❡❧ ❞❡ ✉♠ ♣r♦❢❡ss♦r é ❛❥✉❞❛r ♦ ❛❧✉♥♦ ❛ ✈❡r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ ▼❛t❡♠át✐❝❛ q✉❡ ❡❧❡ ❡st✉❞❛ ❡ ♦✉tr❛s ❝♦✐s❛s✱ ❛✜r♠❛ ❙♣✐♥❡❧❧✐✳ P♦r ❡①❡♠♣❧♦✱ ♥ú♠❡r♦s ❝♦♠♣❧❡①♦s sã♦ ✉s❛❞♦s ♥❛ ❛♥á❧✐s❡ ❞❡ ❝✐r❝✉✐t♦s ❡❧étr✐❝♦s✱ ❧♦❣❛r✐t♠♦s sã♦ ✉s❛❞♦s ♥❛ ❊s❝❛❧❛ ❘✐❝❤t❡r✱ s❡♥♦ ❡ ❝♦ss❡♥♦ ♥♦s ❝á❧❝✉❧♦s ❞❡ r❛♠♣❛s✱ ❡♥tr❡ ♦✉tr❛s✳ ❖ ❛❧✉♥♦ ♥ã♦ ♣♦❞❡ s❡ ❧✐♠✐t❛r ❛ ✈❡r q✉❡ ✉♠ ❛ss✉♥t♦ ❡♠ ▼❛t❡♠át✐❝❛ só s❡r✈❡ ♣❛r❛ ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❡st✉❞♦✳ ❊ ♦✉tr❛✱ q✉❡♠ ❞❡✈❡ ❡st❛❜❡❧❡❝❡r ❛s r❡❧❛çõ❡s ❡♥tr❡ ❛ ▼❛t❡♠át✐❝❛ ❡ ❛s s✐t✉❛çõ❡s ♥❛s q✉❛✐s ❡❧❛ s❡r✈❡ ♥ã♦ é ♦ ♣r♦❢❡ss♦r✱ ❡ s✐♠ ♦ ❛❧✉♥♦✱ ❛✜r♠❛ ♦ ♣r♦❢❡ss♦r ❲❛❧t❡r ❙♣✐♥❡❧❧✐ ❬✶✶❪✱ ♣r♦❢❡ss♦r ❝♦♠ ♠❛✐s ❞❡ ✹✵ ❛♥♦s ❞❡ ❡①♣❡r✐ê♥❝✐❛ ❡♠ s❛❧❛ ❞❡ ❛✉❧❛✳

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