❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❈❡♥tr♦ ❞❡ ❈✐ê♥❝✐❛s ❊①❛t❛s ❡ ❞❛ ◆❛t✉r❡③❛
❉❡♣❛rt❛♠❡♥t♦ ❞❡ ▼❛t❡♠át✐❝❛
▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛
❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚
➪❘❊❆ ❊ ❱❖▲❯▼❊ ❈❖▼ ❖ ❆❯❳❮▲■❖ ❉❖
▼❯▲❚■P▲❆◆❖✱ ❇▲❖❈❖❙ ❈Ú❇■❈❖❙ ❡
❙Ó▲■❉❖❙ ●❊❖▼➱❚❘■❈❖❙
♣♦r
❊❧✐ P❛✉❧♦ ❈❛♠♣♦s
s♦❜ ♦r✐❡♥t❛çã♦ ❞♦
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥✲ t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✲ ✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦✲ ♥❛❧ P❘❖❋▼❆❚✲❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡✲ q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
❈❛t❛❧♦❣❛çã♦ ♥❛ ♣✉❜❧✐❝❛çã♦ ❯♥✐✈❡rs✐❞❛❞❡ ❋❡❞❡r❛❧ ❞❛ P❛r❛í❜❛
❇✐❜❧✐♦t❡❝❛ ❙❡t♦r✐❛❧ ❞♦ ❈❈❊◆
❈✶✾✽❛ ❈❛♠♣♦s✱ ❊❧✐ P❛✉❧♦✳
➪r❡❛ ❡ ✈♦❧✉♠❡ ❝♦♠ ♦ ❛✉①í❧✐♦ ❞♦ ♠✉❧t✐♣❧❛♥♦✱ ❜❧♦❝♦s ❝ú❜✐❝♦s ❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s ✴ ❊❧✐ P❛✉❧♦ ❈❛♠♣♦s✳✕ ❏♦ã♦ P❡ss♦❛✱ ✷✵✶✹✳
✹✼❢✳ ✿ ✐❧✳
❖r✐❡♥t❛❞♦r✿ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉s❛ ❉✐ss❡rt❛çã♦ ✭▼❡str❛❞♦✮ ✲ ❯❋P❇✴❈❈❊◆
✶✳ ▼❛t❡♠át✐❝❛✳ ✷✳ ❈á❧❝✉❧♦ ❞❡ ár❡❛s✳ ✸✳ ❈á❧❝✉❧♦ ❞❡ ✈♦❧✉♠❡s✳ ✹✳ ❋✐❣✉r❛s ♣❧❛♥❛s✳ ✺✳ ❙ó❧✐❞♦s ❣❡♦♠étr✐❝♦s✳
➪❘❊❆ ❊ ❱❖▲❯▼❊ ❈❖▼ ❖ ❆❯❳❮▲■❖ ❉❖
▼❯▲❚■P▲❆◆❖✱ ❇▲❖❈❖❙ ❈Ú❇■❈❖❙ ❡
❙Ó▲■❉❖❙ ●❊❖▼➱❚❘■❈❖❙
♣♦r
❊❧✐ P❛✉❧♦ ❈❛♠♣♦s
❚r❛❜❛❧❤♦ ❞❡ ❈♦♥❝❧✉sã♦ ❞❡ ❈✉rs♦ ❛♣r❡s❡♥t❛❞♦ ❛♦ ❈♦r♣♦ ❉♦❝❡♥t❡ ❞♦ ▼❡str❛❞♦ Pr♦✜ss✐♦♥❛❧ ❡♠ ▼❛t❡♠át✐❝❛ ❡♠ ❘❡❞❡ ◆❛❝✐♦♥❛❧ P❘❖❋▼❆❚ ❈❈❊◆✲❯❋P❇✱ ❝♦♠♦ r❡q✉✐s✐t♦ ♣❛r❝✐❛❧ ♣❛r❛ ♦❜t❡♥çã♦ ❞♦ tít✉❧♦ ❞❡ ▼❡str❡ ❡♠ ▼❛t❡♠át✐❝❛✳
➪r❡❛ ❞❡ ❈♦♥❝❡♥tr❛çã♦✿ ▼❛t❡♠át✐❝❛
❆♣r♦✈❛❞❛ ♣♦r✿
Pr♦❢✳ ❉r✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛ ✲ ❯❋P❇ ✭❖r✐❡♥t❛❞♦r✮
Pr♦❢✳ ❉r✳ ❆❞r✐❛♥♦ ❆❧✈❡s ❞❡ ▼❡❞❡✐r♦s ✲ ❯❋P❇
Pr♦❢✳ ❉r❛✳ ❇ár❜❛r❛ ❈♦st❛ ❞❛ ❙✐❧✈❛ ✲ ❯❋❘P❊
❆❣r❛❞❡❝✐♠❡♥t♦s
❆❣r❛❞❡ç♦ ♣r✐♠❡✐r❛♠❡♥t❡ ❛ ❉❡✉s✱ ♣♦✐s s❡♠ ❡❧❡ ♥❛❞❛ s❡r✐❛ ♣♦ssí✈❡❧✳
❆♦s ♠❡✉s ♣❛✐s✱ ❙❡✈❡r✐♥♦ P❛✉❧♦ ❡ ■r❛❝❡♠❛ ▼❛r✐❛✱ q✉❡ sã♦ ♠✐♥❤❛ ❜❛s❡✱ ♠❡✉s ❡①❡♠♣❧♦s ❞❡ ✈✐❞❛✳
❆ ♠❡✉s ✐r♠ã♦s✱ ❘❛♥❣❡❧ P❛✉❧♦✱ ❘✉t❡ ❡ ❏❛♥❞✐r❛ ❡ à ♠✐♥❤❛ ♥♦✐✈❛✱ ❊❞♥❛ ▼❛r✐❛ ♣❡❧❛ ❝♦♠♣r❡❡♥sã♦ ❡ ♦ ✐♥❝❡♥t✐✈♦ q✉❡ ♠❡ ♣r♦♣♦r❝✐♦♥❛r❛♠✳
❆♦s ♣r♦❢❡ss♦r❡s ❞♦ P❘❖❋▼❆❚✱ ♣❡❧♦ ❡♠♣❡♥❤♦ ❡ ❞❡❞✐❝❛çã♦✱ ❡♠ ❡s♣❡❝✐❛❧ q✉❡r♦ ❛❣r❛❞❡❝❡r ❛♦ ♠❡✉ ♦r✐❡♥t❛❞♦r ♦ Pr♦❢✳ ▼❛♥❛ssés ❳❛✈✐❡r ❞❡ ❙♦✉③❛✱ ♣♦r ❛❥✉❞❛r ♥❛ ❡❧❛❜♦r❛çã♦ ❞♦ ♠❡✉ tr❛❜❛❧❤♦ ❡ ❝♦♥tr✐❜✉✐r ♣❛r❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡st❡ s♦♥❤♦✳
➚ ❇❛♥❝❛ ❝♦♠♣♦st❛ ♣❡❧♦s ♣r♦❢❡ss♦r❡s✿ ❆❞r✐❛♥♦ ❆❧✈❡s ❞❡ ▼❡❞❡✐r♦s ❡ ❇ár❜❛r❛ ❈♦st❛ ❞❛ ❙✐❧✈❛✳
❆♦s ♠❡✉s ❝♦❧❡❣❛s ❞❡ ❝✉rs♦ ♣❡❧♦ ❝♦♥✈í✈✐♦ ❤❛r♠♦♥✐♦s♦ q✉❡ t✐✈❡♠♦s ♥❡ss❡s ❞♦✐s ❛♥♦s ❡ t♦❞❛ tr♦❝❛ ❞❡ ❝♦♥❤❡❝✐♠❡♥t♦s q✉❡ t❛♥t♦ ♠❡ ❡♥r✐q✉❡❝❡r❛♠✳
❆♦s ❝♦❧❡❣❛s ❞❡ ❡st✉❞♦ ❞♦ P❘❖❋▼❆❚✱ ❏♦ã♦ P❛✉❧♦✱ ❘❡♥❛t♦✱ ❊❞❥❛♥❡✱ ❏♦s✐❧❞♦✱ ▼❛r❝♦♥✲ ❞❡s✱ ❉❡♠✐❧s♦♥ ❡ ❆♥tô♥✐♦ ♣❡❧❛ ❛♠✐③❛❞❡✱ ❢♦rç❛ ❡ ✐♥❝❡♥t✐✈♦ ♥♦s ♠♦♠❡♥t♦s ♠❛✐s ❞✐❢í❝❡✐s ❞♦ ❝✉rs♦✳
❆ t♦❞♦s q✉❡ ❢❛③❡♠ ♣❛rt❡ ❞❡ ♠✐♥❤❛ ✈✐❞❛ ❡ q✉❡ ❝♦♥tr✐❜✉ír❛♠ ❞✐r❡t❛✱ ♦✉ ✐♥❞✐r❡t❛♠❡♥t❡✱ ♣❛r❛ q✉❡ ❡✉ ❝❤❡❣❛ss❡ ❛té ❛q✉✐✳
❉❡❞✐❝❛tór✐❛
❘❡s✉♠♦
❖ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s é ❞❡ ❣r❛♥❞❡ ✐♠♣♦rtâ♥❝✐❛ ♥❛ ▼❛t❡♠át✐❝❛ ❡ ♣♦ss✉✐ ✈❛st❛ ❛♣❧✐✲ ❝❛çã♦ ♥♦ ♥♦ss♦ ❝♦t✐❞✐❛♥♦✳ ❖ ♦❜❥❡t✐✈♦ ❞❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ♣r♦❞✉③✐r ✉♠ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ♣❛r❛ ❞❛r ❛✉❧❛s s♦❜r❡ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s ❛♦s ❛❧✉♥♦s ❞♦ 2◦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❞❛ ❊s❝♦❧❛ ❏❛r✐♥❛ ▼❛✐❛✱ ❧♦❝❛❧✐③❛❞❛ ♥♦ ♠✉♥✐❝í♣✐♦
❞❡ ❏♦ã♦ ❆❧❢r❡❞♦✱ P❡r♥❛♠❜✉❝♦✳ ❯♠ t❡st❡ ❞✐❛❣♥óst✐❝♦ ❢♦✐ ❛♣❧✐❝❛❞♦ ❛♥t❡s ❞❛ ♣r❡♣❛r❛çã♦ ❞♦ ♠❛t❡r✐❛❧ ♣❛r❛ ✐❞❡♥t✐✜❝❛r ❛s ❞✐✜❝✉❧❞❛❞❡s ❞♦s ❛❧✉♥♦s ♥❛ r❡s♦❧✉çã♦ ❞❡ q✉❡stõ❡s ❡♥✈♦❧✈❡♥❞♦ ♦s ❝♦♥❝❡✐t♦s ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡✳ ❉❡♣♦✐s ❞❡ tr❛❜❛❧❤❛❞♦s ♦s ❝♦♥t❡ú❞♦s✱ ♦✉tr♦ t❡st❡ ❢♦✐ ❛♣❧✐✲ ❝❛❞♦ ♣❛r❛ ✈❡r✐✜❝❛r s❡ ❤♦✉✈❡ ❛❧❣✉♠ ❛✈❛♥ç♦✳ ❖ ♠❛t❡r✐❛❧ ♣r♦❞✉③✐❞♦ t❡♠ ♦ ♥í✈❡❧ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✱ ♦✉ s❡❥❛✱ ♥ã♦ t❡♠ t♦❞❛ ❛ r✐❣✐❞❡③ ❞❛s ❞❡♠♦♥str❛çõ❡s ❞♦ ❡♥s✐♥♦ s✉♣❡r✐♦r✱ ❛♣r❡s❡♥t❛ ❛♣❡♥❛s ❛❧❣✉♠❛s ❞❡❞✉çõ❡s ❡✴♦✉ ❞❡♠♦♥str❛çõ❡s s✐♠♣❧❡s ♣❛r❛ ❢❛❝✐❧✐t❛r ♦ ♣r♦❝❡ss♦ ❞❡ ❡♥s✐♥♦✲ ❛♣r❡♥❞✐③❛❣❡♠✳ ❚❛♠❜é♠ ✉s❛♠♦s ♠❛t❡r✐❛✐s ❞♦ ❧❛❜♦r❛tór✐♦ ❞❡ ▼❛t❡♠át✐❝❛ ❡♠ ✉♠❛ ♦✜❝✐♥❛ ♣❛r❛ ❡①♣❧✐❝❛r ♦s ❝♦♥❝❡✐t♦s ❡ ❢ór♠✉❧❛s ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡✳
P❛❧❛✈r❛s✲❝❤❛✈❡✿ ➪r❡❛✱ ❱♦❧✉♠❡✱ ✜❣✉r❛s ♣❧❛♥❛s✱ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✳
❆❜str❛❝t
❚❤❡ ❝❛❧❝✉❧❛t✐♦♥ ♦❢ ❛r❡❛s ❛♥❞ ✈♦❧✉♠❡s ✐s ♦❢ ❣r❡❛t ✐♠♣♦rt❛♥❝❡ ✐♥ ▼❛t❤❡♠❛t✐❝s✱ ❛♥❞ ✐s ✇✐❞❡❧② ❛♣♣❧✐❝❛❜❧❡ ✐♥ ♦✉r ❞❛✐❧② ❧✐✈❡s✳ ❚❤❡ ♦❜❥❡❝t✐✈❡ ♦❢ t❤✐s st✉❞② ✇❛s t♦ ♣r♦❞✉❝❡ s✉♣♣♦rt ♠❛t❡r✐❛❧ ❢♦r ❝❧❛ss❡s t❡❛❝❤✐♥❣ ❤♦✇ t♦ ❝❛❧❝✉❧❛t❡ t❤❡ ❛r❡❛s ♦❢ ✢❛t ✜❣✉r❡s ❛♥❞ t❤❡ ✈♦❧✉♠❡s ♦❢ ❣❡♦♠❡tr✐❝ s♦❧✐❞s✳ ❚❤❡s❡ ❝❧❛ss❡s ✇❡r❡ ❣✐✈❡♥ t♦ s❡❝♦♥❞✲②❡❛r ❤✐❣❤ s❝❤♦♦❧ st✉❞❡♥ts ❛t t❤❡ ❏❛r✐♥❛ ▼❛✐❛ ❙❝❤♦♦❧✱ ❧♦❝❛t❡❞ ✐♥ t❤❡ t♦✇♥ ♦❢ ❏♦ã♦ ❆❧❢r❡❞♦✱ P❡r♥❛♠❜✉❝♦✳ ❆ ❞✐❛❣♥♦st✐❝ t❡st ✇❛s ❣✐✈❡♥ ❜❡❢♦r❡ t❤❡ ♠❛t❡r✐❛❧ ✇❛s ♣r❡♣❛r❡❞✱ t♦ ✐❞❡♥t✐❢② st✉❞❡♥ts✬ ❞✐✣❝✉❧t✐❡s ✐♥ s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ✐♥✈♦❧✈✐♥❣ t❤❡ ❝♦♥❝❡♣ts ♦❢ ❛r❡❛ ❛♥❞ ✈♦❧✉♠❡✳ ❆❢t❡r t❤❡ ❝♦♥t❡♥t ✇❛s t❛✉❣❤t✱ ❛♥♦t❤❡r t❡st ✇❛s ❣✐✈❡♥ t♦ ✈❡r✐❢② ✇❤❡t❤❡r t❤❡r❡ ❤❛❞ ❜❡❡♥ ❛♥② ♣r♦❣r❡ss✳ ❚❤❡ ♠❛t❡r✐❛❧ ♣r♦❞✉❝❡❞ ✇❛s ♠❛❞❡ ❢♦r ❤✐❣❤ s❝❤♦♦❧ ❧❡✈❡❧✱ ✐♥ ♦t❤❡r ✇♦r❞s✱ ✐t ✇❛s ♥♦t ❛s r✐❣✐❞ ❛s ❛ ❤✐❣❤❡r ❡❞✉❝❛t✐♦♥ ♣r❡s❡♥t❛t✐♦♥ ✇♦✉❧❞ ❤❛✈❡ ❜❡❡♥❀ ✐t ♦♥❧② ♣r❡s❡♥t❡❞ s♦♠❡ s✐♠♣❧❡ ❞❡❞✉❝t✐♦♥s ❛♥❞✴♦r ❞❡♠♦♥str❛t✐♦♥s t♦ ❢❛❝✐❧✐t❛t❡ t❤❡ t❡❛❝❤✐♥❣✲❧❡❛r♥✐♥❣ ♣r♦❝❡ss✳ ❲❡ ❛❧s♦ ✉s❡❞ ♠❛t❤❡♠❛t✐❝s ❧❛❜♦r❛t♦r② ♠❛t❡r✐❛❧s ✐♥ ❛ ✇♦r❦s❤♦♣ t♦ ❡①♣❧❛✐♥ t❤❡ ❝♦♥❝❡♣ts ❛♥❞ ❢♦r♠✉❧❛s ❢♦r ❛r❡❛ ❛♥❞ ✈♦❧✉♠❡✳
❑❡②✇♦r❞s✿ ❆r❡❛✱ ❱♦❧✉♠❡✱ ♣❧❛♥❡ ✜❣✉r❡s✱ ❣❡♦♠❡tr✐❝ s♦❧✐❞s✳
❙✉♠ár✐♦
✶ ❚❡st❡ ❞✐❛❣♥óst✐❝♦ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✷ ✶✳✶ ◗✉❡stõ❡s s❡❣✉✐❞❛s ❞❡ s✉❛s ❛♥á❧✐s❡s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷ ✶✳✷ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✽ ✶✳✷✳✶ ◗✉❛♥t✐t❛t✐✈♦✴♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛❝❡rt❛r❛♠ ❝❛❞❛ q✉❡stã♦ ✳ ✳ ✳ ✽ ✶✳✷✳✷ ◗✉❛♥t✐❞❛❞❡ ❞❡ ❛❝❡rt♦s ❞♦s ❛❧✉♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✾
✷ ❊st✉❞♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡ ✶✶
✷✳✶ ➪r❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶ ✷✳✶✳✶ ❈♦♥❝❡✐t♦ ❞❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶✳✷ ➪r❡❛ ❞♦ q✉❛❞r❛❞♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷ ✷✳✶✳✸ ➪r❡❛ ❞♦ r❡tâ♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✸ ✷✳✶✳✹ ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✺ ✷✳✶✳✺ ➪r❡❛ ❞♦ tr✐â♥❣✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✻ ✷✳✶✳✻ ➪r❡❛ ❞♦ tr❛♣é③✐♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽ ✷✳✶✳✼ ➪r❡❛ ❞♦ ❧♦s❛♥❣♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾ ✷✳✶✳✽ ➪r❡❛ ❞❡ ✉♠ P♦❧í❣♦♥♦ r❡❣✉❧❛r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✵ ✷✳✶✳✾ ➪r❡❛ ❞♦ ❝ír❝✉❧♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶ ✷✳✶✳✶✵ ❈❛❧❝✉❧❛♥❞♦ ár❡❛s ♣♦r ❛♣r♦①✐♠❛çã♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸ ✷✳✶✳✶✶ ❆t✐✈✐❞❛❞❡s ❝♦♠ ♠❛t❡r✐❛✐s ❝♦♥❝r❡t♦s ❡ ❛❧❣✉♥s ❡①❡r❝í❝✐♦s tr❛❜❛❧❤❛❞♦s
❝♦♠ ♦s ❛❧✉♥♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✹ ✷✳✷ ❱♦❧✉♠❡ ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼ ✷✳✷✳✶ ❯♥✐❞❛❞❡ ♣❛❞rã♦ ❞❡ ✈♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✽ ✷✳✷✳✷ ❘❡❧❛çã♦ ❡♥tr❡ ♠❡❞✐❞❛ ❞❡ ✈♦❧✉♠❡ ❡ ❝❛♣❛❝✐❞❛❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✾ ✷✳✷✳✸ Pr✐♥❝í♣✐♦ ❞❡ ❈❛✈❛❧✐❡r✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✵
✷✳✷✳✹ ❱♦❧✉♠❡ ❞♦ ♣r✐s♠❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✶ ✷✳✷✳✺ ❱♦❧✉♠❡ ❞❛ ♣✐râ♠✐❞❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷ ✷✳✷✳✻ ❱♦❧✉♠❡ ❞♦ ❝✐❧✐♥❞r♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✸ ✷✳✷✳✼ ❱♦❧✉♠❡ ❞♦ ❝♦♥❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✺ ✷✳✷✳✽ ❱♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✻ ✷✳✷✳✾ ❆t✐✈✐❞❛❞❡s ❞❡s❡♥✈♦❧✈✐❞❛s ❝♦♠ ♦s ❛❧✉♥♦s s♦❜r❡ ✈♦❧✉♠❡ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽
✸ ❊①❛♠❡ ❞✐❛❣♥óst✐❝♦ ✜♥❛❧ ❡ ❛♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ✹✵ ✸✳✶ ❚❡st❡ ❞✐❛❣♥óst✐❝♦ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✵ ✸✳✷ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✸ ✸✳✷✳✶ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ♠❛r❝❛r❛♠ ❝❛❞❛ ❛❧t❡r♥❛t✐✈❛ ♣♦r q✉❡stã♦ ✳ ✳ ✹✸ ✸✳✷✳✷ P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✈❡rs✉s ♥ú♠❡r♦ ❞❡ ❛❝❡rt♦s ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✹✺
❘❡❢❡rê♥❝✐❛s ❇✐❜❧✐♦❣rá✜❝❛s ✹✼
■♥tr♦❞✉çã♦
❈❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡ ♦ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s ♥ã♦ é ✉♠❛ ♥❡❝❡ss✐❞❛❞❡ r❡❝❡♥t❡✱ ❞❡s❞❡ ❛s ❝✐✈✐❧✐③❛çõ❡s ❛♥t✐❣❛s ❡ss❡ ♣r♦❜❧❡♠❛ ❛♣❛r❡❝❡✉✱ ❡♠ ♠✉✐t♦s ❝❛s♦s✱ ♣♦r ♥❡❝❡ss✐❞❛❞❡s ❞♦ ❝♦t✐❞✐❛♥♦✱ ❝♦♠♦ ❛♣❧✐❝❛çã♦ ♥❛ ❛❣r✐❝✉❧t✉r❛✱ ♥❛s ❝♦♥str✉çõ❡s✱ ❡t❝✳ ❡ ❡♠ ♦✉tr♦s ❛♣❡♥❛s ♣❡❧❛ ❜❡❧❡③❛ q✉❡ ❛ ▼❛t❡♠át✐❝❛ ♣♦❞❡ ♣r♦♣♦r❝✐♦♥❛r✳ ❊ss❛s ♥❡❝❡ss✐❞❛❞❡s ❧❡✈❛r❛♠ ♦ ❤♦♠❡♠ ❛ ❞❡s❡♥✈♦❧✈❡r ✐❞❡✐❛s ❡ ♠ét♦❞♦s ♣❛r❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ss❡ t✐♣♦ ❞❡ ❝á❧❝✉❧♦✳ ❉❡s❞❡ ♦ ❊♥s✐♥♦ ❋✉♥❞❛♠❡♥t❛❧✱ ♦s ❛❧✉♥♦s ❥á ❡st✉❞❛♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡✳ ▼❛s ♥❛ ♠❛✐♦r✐❛ ❞❛s ✈❡③❡s ❛♣r❡s❡♥t❛♠♦s ❛♣❡♥❛s ❛s ❢ór♠✉❧❛s ❡ ♣❡❞✐♠♦s ♣r❛ ❡❧❡s r❡s♦❧✈❡r❡♠ ❛s q✉❡stõ❡s ♣r♦♣♦st❛s✳ ❆♦ ❧♦♥❣♦ ❞♦s ❛♥♦s ❡♥s✐♥❛♥❞♦ ❡ss❡ ❝♦♥t❡ú❞♦✱ ♣❡r❝❡❜❡♠♦s ❛s ❞✐✜✲ ❝✉❧❞❛❞❡s q✉❡ ♠✉✐t♦s ❛❧✉♥♦s ❛♣r❡s❡♥t❛♠ ♥❛ ❝♦♠♣r❡❡♥sã♦ ❞❡ss❡s ❝♦♥❝❡✐t♦s✳ ▼✉✐t♦s ❧✐✈r♦s ❞✐❞át✐❝♦s ❞♦ ❡♥s✐♥♦ ❜ás✐❝♦ ❡ ♣r♦❢❡ss♦r❡s ❛♣r❡s❡♥t❛♠ ❛♣❡♥❛s ❛s ❢ór♠✉❧❛s s❡♠ ♥❡♥❤✉♠❛ ❞❡✲ ♠♦♥str❛çã♦ ♦✉ ♣❡❧♦ ♠❡♥♦s ✉♠❛ ❞❡❞✉çã♦ ❞❛s ♠❡s♠❛s✱ ❝♦♠ ✐ss♦ ❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s ❢❛③ ♦s ❝á❧❝✉❧♦s ❞❡ ❢♦r♠❛ ♠❡❝â♥✐❝❛✳ ❚❛❧✈❡③ ♣♦r ❝❛✉s❛ ❞✐ss♦ ❝♦♥❢✉♥❞❛♠ t❛♥t♦✱ ♦s ❝♦♥❝❡✐t♦s ❞❡ ♣❡rí♠❡tr♦✱ ár❡❛ ❡ ✈♦❧✉♠❡✳
◆❡st❡ tr❛❜❛❧❤♦ ♣r❡♣❛r❛♠♦s ✉♠ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ✭❈❛♣ít✉❧♦ ✷✮ ♣❛r❛ ♦ ❡st✉❞♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡ ✈♦❧✉♠❡ ❞❡ ❛❧❣✉♥s só❧✐❞♦s ❣❡♦♠étr✐❝♦s ♣❛r❛ ❛❧✉♥♦s ❞♦ s❡❣✉♥❞♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦✳ ❆♥t❡s ❞❡ ♣r♦❞✉③✐r ♦ ♠❛t❡r✐❛❧ ❛♣❧✐❝❛♠♦s ✉♠ t❡st❡ ❞✐❛❣♥óst✐❝♦ ❝♦♠ ❡①❡r❝í❝✐♦s s♦❜r❡ ár❡❛ ❡ ✈♦❧✉♠❡✳ ❉❡♣♦✐s ❛♥❛❧✐s❛♠♦s ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛s ❞✐✜❝✉❧❞❛❞❡s ❡ ❡rr♦s ❞♦s ❛❧✉♥♦s ♥❛ r❡s♦❧✉çã♦ ❞❛s q✉❡stõ❡s ❡ ❝♦♠ ❜❛s❡ ♥❡ss❡s ❞❛❞♦s ❡❧❛❜♦r❛♠♦s ♦ ♠❛t❡r✐❛❧✳
❆ ♠❛♥❡✐r❛ ❝♦♠♦ ❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ❧❡♠❜r❛ ❛ ♠❡t♦❞♦❧♦❣✐❛ ❞❛ ❊♥❣❡♥❤❛r✐❛ ❉✐❞át✐❝❛ q✉❡ ❡♠❡r❣✐✉ ♥♦ ✐♥í❝✐♦ ❞❛ ❞é❝❛❞❛ ❞❡ ✽✵✱ ❝r✐❛❞❛ ♣♦r ▼✐❝❤è❧❡ ❆rt✐❣✉❡✳ ❙❡❣✉♥❞♦ ❆rt✐❣✉❡ ✭✶✾✾✻✱ ♣✳ ✶✾✻✮✱ ❊♥❣❡♥❤❛r✐❛ ❉✐❞át✐❝❛ s❡ ❝❛r❛❝t❡r✐③❛ ♣♦r ✉♠ ❡sq✉❡♠❛ ❡①♣❡r✐♠❡♥t❛❧ ❜❛s❡❛❞♦ ❡♠ r❡❛❧✐③❛çõ❡s ❞✐❞át✐❝❛s ♥❛ s❛❧❛ ❞❡ ❛✉❧❛✱ ✐st♦ é✱ ♥❛ ❝♦♥❝❡♣çã♦✱ ♥❛ r❡❛❧✐③❛çã♦✱ ♥❛ ♦❜s❡r✈❛çã♦ ❡ ♥❛ ❛♥á❧✐s❡ ❞❡ s❡q✉ê♥❝✐❛s ❞❡ ❡♥s✐♥♦✳ ❆ s❡q✉ê♥❝✐❛ ❞❡st❡ tr❛❜❛❧❤♦ ❢♦✐ ❛♣❧✐❝❛❞❛ s❡❣✉✐♥❞♦ ❡st❛ ♠❡t♦❞♦❧♦❣✐❛✿ ❢♦r❛♠ ❢❡✐t❛s ❛♥á❧✐s❡s ❛ ♣r✐♦r✐ ❞♦s ❝♦♥❤❡❝✐♠❡♥t♦s ♣ré✈✐♦s ❞♦s ❛❧✉♥♦s✱ ❝♦♠ ❜❛s❡ ♥♦ t❡st❡ ❞✐❛❣♥óst✐❝♦ ✐♥✐❝✐❛❧✱ ❡♠ s❡❣✉✐❞❛ ❢♦✐ ❞❡s❡♥✈♦❧✈✐❞♦ ♦ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ❡ ♣♦r ✜♠ ❢♦r❛♠ ❢❡✐t❛s ❛♥á❧✐s❡s ❞♦s r❡s✉❧t❛❞♦s ♥♦ t❡st❡ ❞✐❛❣♥óst✐❝♦ ✜♥❛❧ ❝♦♠♣❛r❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s✳
❆❧é♠ ❞♦ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ♣❛r❛ ❛s ❛✉❧❛s✱ ✜③❡♠♦s ✉♠❛ ♦✜❝✐♥❛ ❝♦♠ ✈ár✐❛s ❛t✐✈✐❞❛✲ ❞❡s ✉t✐❧✐③❛♥❞♦ ♠❛t❡r✐❛✐s ❞♦ ▲❛❜♦r❛tór✐♦ ❞❡ ▼❛t❡♠át✐❝❛ ♣❛r❛ ❢❛❝✐❧✐t❛r ❛ ❝♦♠♣r❡❡♥sã♦ ❞♦s ❝♦♥❝❡✐t♦s ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡✳
❈❛♣ít✉❧♦ ✶
❚❡st❡ ❞✐❛❣♥óst✐❝♦ ❡ ❛♥á❧✐s❡ ❞♦s
r❡s✉❧t❛❞♦s
❖ t❡st❡ ❞✐❛❣♥óst✐❝♦ ❢♦✐ ❛♣❧✐❝❛❞♦ ♣❛r❛ ✶✵✼ ❛❧✉♥♦s ❞❡ três t✉r♠❛s ❞♦ ✷♦ ❛♥♦ ❞♦ ❡♥s✐♥♦
♠é❞✐♦ ❞❛ ❊s❝♦❧❛ ❊st❛❞✉❛❧ ❏❛r✐♥❛ ▼❛✐❛✱ ❧♦❝❛❧✐③❛❞❛ ♥♦ ♠✉♥✐❝í♣✐♦ ❞❡ ❏♦ã♦ ❆❧❢r❡❞♦✱ P❡r♥❛♠✲ ❜✉❝♦✳ ❖ ♦❜❥❡t✐✈♦ ❞♦ t❡st❡ ❡r❛ ✈❡r✐✜❝❛r q✉❛✐s ❡r❛♠ ❛s ❞✐✜❝✉❧❞❛❞❡s ❞♦s ❛❧✉♥♦s ❡♠ r❡s♦❧✈❡r ♣r♦❜❧❡♠❛s q✉❡ ❡♥✈♦❧✈✐❛♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡ ♦ ✈♦❧✉♠❡ ❞❡ ❛❧❣✉♥s só✲ ❧✐❞♦s ❣❡♦♠étr✐❝♦s✳ ❖ t❡st❡ ❢♦✐ ❝♦♠♣♦st♦ ♣♦r q✉✐♥③❡ q✉❡stõ❡s✱ s❡♥❞♦ ♥♦✈❡ s♦❜r❡ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡ s❡✐s s♦❜r❡ ✈♦❧✉♠❡✳ Pr✐♠❡✐r♦ ❛♥❛❧✐s❛♠♦s ❛s r❡s♣♦st❛s ❞♦s ❛❧✉♥♦s ❡♠ ❝❛❞❛ q✉❡stã♦✳ ❆❜❛✐①♦ t❡♠♦s ♦ t❡st❡ ❡ ❛♦ ❧❛❞♦ ❞❛s ❛❧t❡r♥❛t✐✈❛s t❡♠♦s ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛ ♠❛r❝❛r❛♠✳
✶✳✶ ◗✉❡stõ❡s s❡❣✉✐❞❛s ❞❡ s✉❛s ❛♥á❧✐s❡s
◗✉❡stã♦ 1✲ ❙❛❜❡♥❞♦ q✉❡ ❛ ✜❣✉r❛ ❛❜❛✐①♦ é ❢♦r♠❛❞❛ ♣♦r 25 q✉❛❞r❛❞✐♥❤♦s ✐❣✉❛✐s ❡ q✉❡ ❛
♠❡❞✐❞❛ ❞❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ❤❛❝❤✉r❛❞♦ é ✐❣✉❛❧ ❛1cm2✳ ◗✉❛❧ é ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ t♦t❛❧ ❞❛
✜❣✉r❛❄
(a) 25cm2 37% (b) 30cm2 1% (c) 35cm2 0% (d) 40cm2 0% (e) 50cm2 62%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❊
❖ ♦❜❥❡t✐✈♦ ❞❡ss❛ q✉❡stã♦ ❡r❛ ✈❡r✐✜❝❛r s❡ ♦s ❛❧✉♥♦s ❝♦♠♣r❡❡♥❞✐❛♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✳ ❊ss❛ q✉❡stã♦ t❡♠ ✉♠ ♥í✈❡❧ ❜❡♠ s✐♠♣❧❡s ♣❛r❛ ❛❧✉♥♦s ❞♦ 2♦ ❛♥♦ ❞♦
❡♥s✐♥♦ ♠é❞✐♦✱ ❡ ♠❡s♠♦ ❛ss✐♠ s♦♠❡♥t❡62% ♠❛r❝❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛E✱ q✉❡ é ❛ ❛❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✱ ♦ q✉❡ é ✉♠ ♣❡r❝❡♥t✉❛❧ ❜❛✐①♦ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ♥í✈❡❧ ❞❛ q✉❡stã♦ ❡ ❛ sér✐❡ ❞♦s ❛❧✉♥♦s✳ ❖✉tr♦ ❢❛t♦ ✐♠♣♦rt❛♥t❡ é q✉❡ 37% ❞♦s ❛❧✉♥♦s ♠❛r❝❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ A✱
♦✉ s❡❥❛✱ ❝♦♥t❛r❛♠ ❛♣❡♥❛s ♦ ♥ú♠❡r♦ ❞❡ q✉❛❞r❛❞♦s ❡ ♥ã♦ ♣❡r❝❡❜❡r❛♠ q✉❡ ❛ ár❡❛ ❞❡ ❝❛❞❛ q✉❛❞r❛❞♦ ❡q✉✐✈❛❧❡ à ár❡❛ ❞❡ ❞♦✐s tr✐â♥❣✉❧♦s✳
✶✳✶✳ ◗❯❊❙❚Õ❊❙ ❙❊●❯■❉❆❙ ❉❊ ❙❯❆❙ ❆◆➪▲■❙❊❙
◗✉❡stã♦ 2 ✲ ❯♠ ♣❡❞r❡✐r♦ ♣r❡❝✐s❛ r❡✈❡st✐r ♦ ♣✐s♦ ❞❡ ✉♠❛ s❛❧❛ r❡t❛♥❣✉❧❛r ❞❡ 9m ♣♦r 7m ❝♦♠ ❝❡râ♠✐❝❛✳ ◗✉❛♥t♦s ♠❡tr♦s q✉❛❞r❛❞♦s ❞❡ ❝❡râ♠✐❝❛ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ r❡✈❡st✐r ♦ ♣✐s♦ ❞❡ss❛ s❛❧❛❄
(a) 16m2 10% (b) 63m2 57% (c) 32m2 23% (d) 64m2 5% (e) 126m2 5%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❇
❊ss❛ q✉❡stã♦ t❛♠❜é♠ ❞❡ ♥í✈❡❧ ❢á❝✐❧✱ ♣r♦❝✉r❛✈❛ ✈❡r✐✜❝❛r s❡ ♦s ❛❧✉♥♦s s❛❜✐❛♠ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ ✉♠ r❡tâ♥❣✉❧♦✳ 57% ❞❡❧❡s ♠❛r❝❛r❛♠ ❛ ❛❧t❡r♥❛t✐✈❛ ❇ q✉❡ é ❛ ❛ ❛❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✳
▼❛s ♦ q✉❡ ❝❤❛♠❛ ❛t❡♥çã♦ é q✉❡ 23% ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❈ q✉❡ é ♦ ♣❡rí♠❡tr♦ ❞❛ ✜❣✉r❛ ❡
♦✉tr♦s10%♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❆✱ q✉❡ é ♦ s❡♠✐♣❡rí♠❡tr♦ ♦✉ ❛ s♦♠❛ ❞♦s ♥ú♠❡r♦s ♠♦str❛❞♦s
♥❛ ✜❣✉r❛✳
◗✉❡stã♦ ✸ ✲ ❯♠❛ ❢❡st❛ ❢♦✐ r❡❛❧✐③❛❞❛ ♥✉♠ ❡s♣❛ç♦ ❞❡ ❢♦r♠❛t♦ r❡t❛♥❣✉❧❛r ❞❡220m♣♦r50m✳
❙❛❜❡♥❞♦ q✉❡ ❡♠ ❝❛❞❛m2 ❤❛✈✐❛ ❡♠ ♠é❞✐❛ ✹ ♣❡ss♦❛s✱ ♣♦❞❡♠♦s ❛✜r♠❛r q✉❡ ♥❛ ❢❡st❛ ❤❛✈✐❛✿
✭❛✮ 274 ♣❡ss♦❛s 11%
✭❜✮ 11004 ♣❡ss♦❛s 14%
✭❝✮ 44000 ♣❡ss♦❛s 28%
✭❞✮ 2750 ♣❡ss♦❛s 24%
✭❡✮ 11000 ♣❡ss♦❛s 23%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈
P❛r❛ r❡s♣♦♥❞❡r ❡ss❛ q✉❡stã♦ ❝♦rr❡t❛♠❡♥t❡✱ ♦ ❛❧✉♥♦ ♣r❡❝✐s❛✈❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❝♦rr❡✲ t❛♠❡♥t❡ ❡ ❞❡♣♦✐s ♣❡r❝❡❜❡r q✉❡ ❡♠ ❝❛❞❛ ♠❡tr♦ q✉❛❞r❛❞♦ ❤❛✈✐❛ ❡♠ ♠é❞✐❛ q✉❛tr♦ ♣❡ss♦❛s✱ ♦✉ s❡❥❛✱ ❞❡✈❡r✐❛ ♠✉❧t✐♣❧✐❝❛r ❛ ár❡❛ ♣♦r 4✳ ❙♦♠❡♥t❡ 28% ❛❝❡rt❛r❛♠ ❡ss❛ q✉❡stã♦✳ ❖✉tr♦s 24% ♠❛r❝❛r❛♠ ❧❡tr❛ ❉✱ ♦✉ s❡❥❛✱ ❞✐✈✐❞✐r❛♠ ❛ ár❡❛ ♣♦r 4❀ ❡23% ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❊✱ ❡ss❡s
❝♦♥s✐❞❡r❛r❛♠ ❛♣❡♥❛s ❛ ár❡❛ ❞❛ ✜❣✉r❛✳ ❖s 14% q✉❡ ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❇ s♦♠❛r❛♠ ❛ ár❡❛
❝♦♠ 4 ❡ ♦s 11% q✉❡ ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❆✱ ❛♣❡♥❛s s♦♠❛r❛♠ ♦s ❞❛❞♦s ❞❛ q✉❡stã♦✳
◗✉❡stã♦ ✹ ✲ ❆ ✜❣✉r❛ ❛❜❛✐①♦ r❡♣r❡s❡♥t❛ s❡t❡ ❤❡①á❣♦♥♦s r❡❣✉❧❛r❡s ♠❡♥♦r❡s ✐❣✉❛✐s ❡ ✉♠ ❤❡①á❣♦♥♦ ♠❛✐♦r✱ ❝✉❥♦s ✈ért✐❝❡s ❝♦✐♥❝✐❞❡♠ ❝♦♠ ♦s ❝❡♥tr♦s ❞❡ s❡✐s ❞♦s ❤❡①á❣♦♥♦s ♠❡♥♦r❡s✳ ❙❛❜❡♥❞♦ q✉❡ ❛ ár❡❛ ❞❛ ✜❣✉r❛ ❤❛❝❤✉r❛❞❛ é2cm2 ✱ q✉❛❧ é ❛ ár❡❛ t♦t❛❧ ❞❛ ✜❣✉r❛ ❡♠ cm2 ❄
(a) 42 35% (b) 6 12% (c) 14 30% (d) 16 21% (e) 50 2%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❆
✶✳✶✳ ◗❯❊❙❚Õ❊❙ ❙❊●❯■❉❆❙ ❉❊ ❙❯❆❙ ❆◆➪▲■❙❊❙
◆❡ss❛ q✉❡stã♦✱ ♦ ❛❧✉♥♦ ❞❡✈❡r✐❛ ♣❡r❝❡❜❡r q✉❡ ❝❛❞❛ ♣❛rt❡ ❤❛❝❤✉r❛❞❛ ❞❛ ✜❣✉r❛ t✐♥❤❛ ár❡❛ 2cm2 ❡ ❞❡♣♦✐s ✈❡r q✉❡ q✉❛♥t✐❞❛❞❡ ❞❡ ✜❣✉r❛s ♣✐♥t❛❞❛s ❝♦❜r✐❛ t♦❞❛ ❛ ✜❣✉r❛ ♠❛✐♦r
❢♦r♠❛❞❛ ♣❡❧♦s s❡t❡ ❤❡①á❣♦♥♦s✳ ❙♦♠❡♥t❡ 35% ❛❝❡rt❛r❛♠✳ ❖✉tr♦s 30% ♠❛r❝❛r❛♠ ❧❡tr❛ ❈
❡ 21% ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❉✱ ♦✉ s❡❥❛✱ ❝♦♥s✐❞❡r❛r❛♠ q✉❡ ❝❛❞❛ ❤❡①á❣♦♥♦ ❡r❛ ❢♦r♠❛❞♦ ♣♦r
❛♣❡♥❛s ❞✉❛s ✜❣✉r❛s ♣✐♥t❛❞❛s ❡ ♥❛ ❧❡tr❛ ❈ ❛✐♥❞❛ ❝♦♥s✐❞❡r❛r❛♠ ♦s ♦✐t♦s ❤❡①á❣♦♥♦s✳
◗✉❡stã♦ ✺ ✲ ❯♠❛ ❡s❝♦❧❛ t❡♠ s❡✉s ❝❛♥t❡✐r♦s ❡♠ ❢♦r♠❛ ❣❡♦♠étr✐❝❛✳ ❯♠ ❞❡❧❡s t❡♠ ♦ ❢♦r♠❛t♦ ❞♦ tr❛♣é③✐♦ r❡tâ♥❣✉❧♦✱ ❝♦♠ ❛s ♠❡❞✐❞❛s ✐♥❞✐❝❛❞❛s ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳ ❆ ár❡❛ ❞♦ ❝❛♥t❡✐r♦ r❡♣r❡s❡♥t❛❞❛ ♣❡❧❛ ✜❣✉r❛ é✿
(a) 40m2 30% (b) 30m2 17% (c) 25m2 23% (d) 35m2 14% (e) 45m2 16%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❉
❖ ♦❜❥❡t✐✈♦ ❞❛ q✉❡stã♦ ❡r❛ ✈❡r✐✜❝❛r s❡ ♦ ❛❧✉♥♦ s❛❜✐❛ ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞❡ ✉♠ tr❛♣é✲ ③✐♦✳ ▼❛s s♦♠❡♥t❡ ✶✹✪ ❞❡❧❡s r❡s♣♦♥❞❡r❛♠✲♥❛ ❝♦rr❡t❛♠❡♥t❡✳ ❖ ♠❛✐♦r ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✭✸✵✪✮ ♠❛r❝♦✉ ❧❡tr❛ ❆✱ ♦✉ s❡❥❛✱ ♠✉❧t✐♣❧✐❝♦✉ ❛♣❡♥❛s ❛ ❜❛s❡ ♠❛✐♦r ♣❡❧❛ ❛❧t✉r❛✱ ♦✉tr♦s ♠❛r✲ ❝❛r❛♠ ❛s ❧❡tr❛s ❇✱ ❈ ♦✉ ❊ q✉❡ ♥ã♦ t✐♥❤❛♠ q✉❛❧q✉❡r ❧ó❣✐❝❛✳ ■ss♦ ♥♦s ❧❡✈❛ ❛ ❝♦♥❝❧✉sã♦ ❞❡ q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s ♥ã♦ ❝♦♠♣r❡❡♥❞❡♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛✱ ♣♦✐s s❡ ❡❧❡s ♥ã♦ s♦✉❜❡s✲ s❡♠ ❛ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞♦ tr❛♣é③✐♦ ♣♦❞❡r✐❛♠ ❞❡❝♦♠♣♦r ❡ss❛ ✜❣✉r❛ ❡♠ ✉♠ tr✐â♥❣✉❧♦ ❡ ✉♠ r❡tâ♥❣✉❧♦ ❡ ❝❛❧❝✉❧❛r s✉❛s ár❡❛s ♦❜t❡♥❞♦ ❝♦♠ ✐ss♦ ♦ ♠❡s♠♦ r❡s✉❧t❛❞♦✳
◗✉❡stã♦ ✻ ✲ ❆ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❛ ✜❣✉r❛ ❛❜❛✐①♦ é✿
(a) 24cm2 3% (b) 30cm2 6% (c) 33cm2 8%
(d) 26cm2 73% (e) 48cm2 10%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❇
◆❡ss❛ q✉❡stã♦✱ ♦ ❛❧✉♥♦ ❞❡✈❡r✐❛ ❞❡❝♦♠♣♦r ❛ ✜❣✉r❛ ❡♠ ♣♦❧í❣♦♥♦s ❝♦♥❤❡❝✐❞♦s✱ ❝❛❧❝✉❧❛r s✉❛s ár❡❛s ❡ ❞❡♣♦✐s s♦♠á✲❧❛s✳ ▼❛s ❛♣❡♥❛s ✻✪ ❝♦♥s❡❣✉✐r❛♠ ♣❡r❝❡❜❡r ✐ss♦✳ ❋♦✐ ❛ q✉❡stã♦ ❝♦♠ ♦ ♠❡♥♦r ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦✳ ❆ ♠❛✐♦r✐❛ ❛❜s♦❧✉t❛ ✭✼✸✪✮ ♠❛r❝♦✉ ❛ ❧❡tr❛ ❉✱ q✉❡ é ❛ ♠❡❞✐❞❛ ❞♦ ♣❡rí♠❡tr♦ ❞❛ ✜❣✉r❛✳ ▼❡s♠♦ ♥♦ ❡♥s✐♥♦ ♠é❞✐♦ ♠✉✐t♦s ❛❧✉♥♦s ❛✐♥❞❛ ❝♦♥❢✉♥❞❡♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣❡rí♠❡tr♦ ❝♦♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛✳
✶✳✶✳ ◗❯❊❙❚Õ❊❙ ❙❊●❯■❉❆❙ ❉❊ ❙❯❆❙ ❆◆➪▲■❙❊❙
◗✉❡stã♦ ✼ ✲ ◗✉❛❧ ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❝♦rr❡s♣♦♥❞❡♥t❡ à ♣❛rt❡ ♣✐♥t❛❞❛ ❞❛ ✜❣✉r❛❄
(a) 24cm2 33% (b) 28cm2 9%
(c) 40cm2 9% (d) 48cm2 31% (e) 96cm2 18%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❆
❆s ❛❧t❡r♥❛t✐✈❛s ❆ ✭❝♦rr❡t❛✮ ❡ ❉ ✜❝❛r❛♠ ❜❡♠ ❡q✉✐❧✐❜r❛❞❛s ❝♦♠ ✸✸✪ ❡ ✸✶✪✱ ♦✉ s❡❥❛✱ ♠✉✐t♦s ❛❧✉♥♦s ♠✉❧t✐♣❧✐❝❛r❛♠ 6 ♣♦r 8✱ ♠❛s ♥ã♦ ❞✐✈✐❞✐r❛♠ ♣♦r 2 ♣❛r❛ ❡♥❝♦♥tr❛r ❛♣❡♥❛s ❛
ár❡❛ ❞♦ tr✐â♥❣✉❧♦✳ ❖✉tr♦s ✶✽✪ ❛✐♥❞❛ ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❊✱ ❝❛❧❝✉❧❛♥❞♦ ❝♦♠ ✐ss♦ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦✳
◗✉❡stã♦ ✽ ✲ ◆❛ ✜❣✉r❛ ❛❜❛✐①♦✱ ♦ q✉❛❞r❛❞♦ ❡①t❡r♥♦ t❡♠ ❧❛❞♦ ♠❡❞✐♥❞♦ 7cm✱ ❡♥q✉❛♥t♦ ♦ ❧❛❞♦ ❞♦ q✉❛❞r❛❞♦ ♠❡♥♦r ♠❡❞❡4cm✳ ◗✉❛❧ é ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❛ ♣❛rt❡ ♣✐♥t❛❞❛ ❞❛ ✜❣✉r❛❄ (a) 33cm2 15%
(b) 28cm2 57% (c) 11cm2 10% (d) 12cm2 8% (e) 44cm2 10%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❆
◆❡ss❛ q✉❡stã♦✱ ♦ ❛❧✉♥♦ ❞❡✈❡r✐❛ ❝❛❧❝✉❧❛r ❛s ár❡❛s ❞♦s q✉❛❞r❛❞♦s✱ s✉❜tr❛✐r ❛ ♠❡♥♦r ❞❛ ♠❛✐♦r ♣❛r❛ ❡♥❝♦♥tr❛r ❛ ár❡❛ ❞❛ ♣❛rt❡ ❤❛❝❤✉r❛❞❛✱ ♠❛s ❛♣❡♥❛s ✶✺✪ ✜③❡r❛♠ ✐ss♦ ♠❛r❝❛♥❞♦ ❛ ❧❡tr❛ ❆✳ ❆ ♠❛✐♦r✐❛ ✭✺✼✪✮ ♠❛r❝♦✉ ❛ ❧❡tr❛ ❇✱ ♦✉ s❡❥❛✱ ❛♣❡♥❛s ♠✉❧t✐♣❧✐❝❛r❛♠7cm ♣♦r ✹❝♠ ❡ ✶✵✪ s♦♠❛r❛♠ 7cm ❝♦♠ 4cm✳
◗✉❡stã♦ ✾ ✲ ❯♠ ❡♥❣❡♥❤❡✐r♦ ❞❡s❡❥❛ ❝♦♥str✉✐r ✉♠❛ ♣r❛ç❛ ❝♦♠ ❛ ❢♦r♠❛ ❞❡ ✉♠ ❝ír❝✉❧♦✳ ❙❛❜❡♥❞♦ q✉❡ ❛ ♣r❛ç❛ t❡♠ ♦ ❞✐â♠❡tr♦ ❞❡ 40m✱ q✉❛❧ é s✉❛ ár❡❛❄ ❈♦♥s✐❞❡r❡π = 3,14✳ (a) 314m2 15%
(b) 1600m2 26% (c) 1256m2 47% (d) 2512m2 6% (e) 1570m2 6%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈
◗✉❛s❡ ♠❡t❛❞❡ ✭✹✼✪✮ ❞♦s ❛❧✉♥♦s ❛❝❡rt♦✉ ❡ss❛ q✉❡stã♦✱ ✐ss♦ ✐♥❞✐❝❛ q✉❡ ❡❧❡s t❛❧✈❡③ t❡♥❤❛♠ ❞❡❝♦r❛❞♦ ❛ ❢ór♠✉❧❛ ❞❛ ár❡❛ ❞♦ ❝ír❝✉❧♦✱ ♣♦✐s ❡rr❛r❛♠ q✉❡stõ❡s ♠❛✐s s✐♠♣❧❡s ❡
✶✳✶✳ ◗❯❊❙❚Õ❊❙ ❙❊●❯■❉❆❙ ❉❊ ❙❯❆❙ ❆◆➪▲■❙❊❙
❛❝❡rt❛r❛♠ ✉♠❛ q✉❡stã♦ ❞❡ ✉♠ ♥í✈❡❧ ♠✉✐t♦ ♠❛✐s ❞✐❢í❝✐❧✳ ■ss♦ ♠♦str❛ ♠❛✐s ✉♠❛ ✈❡③ q✉❡ ❛ ♠❛✐♦r✐❛ ❛♣❡♥❛s ❞❡❝♦r❛ ❢ór♠✉❧❛s✱ ♠❛s ♥ã♦ ❝♦♠♣r❡❡♥❞❡ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛✳ ✷✻✪ ♠❛r❝❛r❛♠ ❧❡tr❛ ❇✱ q✉❡ é ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❝✐r❝✉♥s❝r✐t♦ ❛♦ ❝ír❝✉❧♦ ❡ ♠✉✐t♦s ❞❡❧❡s ✜③❡r❛♠ ❛té ♦ ❞❡s❡♥❤♦ ♠♦str❛♥❞♦ ✐ss♦✱ ♠❛s ♥ã♦ ♣❡r❝❡❜❡r❛♠ q✉❡ ❛ ár❡❛ ❞♦ ❝ír❝✉❧♦ ❡r❛ ♠❡♥♦r q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦✳
◗✉❡stã♦ ✶✵ ✲ ❙❛❜❡♥❞♦ q✉❡ ❝❛❞❛ ❝✉❜✐♥❤♦ ❞❛ ✜❣✉r❛ ❛❜❛✐①♦ t❡♠ 1cm3 ❞❡ ✈♦❧✉♠❡✱ q✉❛❧ ♦
✈♦❧✉♠❡ ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❛❜❛✐①♦❄
(a) 7cm3 24% (b) 6cm3 8% (c) 4cm3 10% (d) 8cm3 2% (e) 12cm3 56%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❊
▼❡s♠♦ ❛ q✉❡stã♦ s❡♥❞♦ ❞❡ ♥í✈❡❧ ♠✉✐t♦ ❢á❝✐❧ ♣❛r❛ ❛ sér✐❡✱ ❛✐♥❞❛ ❛ss✐♠ ✹✹✪ ❞♦s ❛❧✉♥♦s ❡rr❛r❛♠✱ s❡♥❞♦ q✉❡ ✷✹✪ s♦♠❛r❛♠ ❛♣❡♥❛s ♦s ✈❛❧♦r❡s q✉❡ ❛♣❛r❡❝✐❛♠ ♥❛ ✜❣✉r❛ ❞❡s❡♥❤❛❞❛✳
◗✉❡stã♦ ✶✶ ✲ ◗✉❛❧ ❛ ♠❡❞✐❞❛ ❞♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✉❜♦ ❝✉❥❛ ❛r❡st❛ ♠❡❞❡5cm❄ (a) 15cm3 21%
(b) 25cm3 22%
(c) 30cm3 8% (d) 125cm3 47% (e) 150cm3 2%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❉
◗✉❛s❡ ♠❡t❛❞❡ ❞♦s ❛❧✉♥♦s ✭✹✼✪✮ ❛❝❡rt♦✉ ❛ q✉❡stã♦✱ t❛❧✈❡③ ♣♦rq✉❡ ❜❛st❛✈❛ ♠✉❧t✐♣❧✐❝❛r ♦s ❧❛❞♦s ❞♦ ❝✉❜♦ q✉❡ ❥á ❡st❛✈❛ ❞❡s❡♥❤❛❞♦✳ ▼❡s♠♦ ❝♦♠ t♦❞❛ ❡ss❛ ❢❛❝✐❧✐❞❛❞❡ ✺✸✪ ❡rr❛r❛♠ ❛ q✉❡stã♦✱ ♠❛r❝❛♥❞♦ ♣r✐♥❝✐♣❛❧♠❡♥t❡ ❛s ❧❡tr❛s ❆ ✭✷✶✪✮ ❡ ❇ ✭✷✷✪✮✳ ◆♦ ❝❛s♦ ❞❛ ❧❡tr❛ ❆✱ s♦♠❛r❛♠ ♦s ♥ú♠❡r♦s q✉❡ ❛♣❛r❡❝✐❛♠ ♥❛ ✜❣✉r❛ r❡♣r❡s❡♥t❛❞❛ ♥♦ ❞❡s❡♥❤♦✳ ◆❛ ❧❡tr❛ ❇✱ ♠✉❧t✐♣❧✐❝❛r❛♠ ❛♣❡♥❛s ♦s ✈❛❧♦r❡s ❞❛ ❜❛s❡✳
◗✉❡stã♦ ✶✷ ✲ ❯♠❛ ❝❛✐①❛ ❞✁á❣✉❛✱ ❡♠ ❢♦r♠❛ ❞❡ ♣❛r❛❧❡❧❡♣í♣❡❞♦ r❡tâ♥❣✉❧♦✱ ❞❡ ❞✐♠❡♥sã♦6m✱
✶✳✶✳ ◗❯❊❙❚Õ❊❙ ❙❊●❯■❉❆❙ ❉❊ ❙❯❆❙ ❆◆➪▲■❙❊❙
2m ❡3m t❡♠ ❝❛♣❛❝✐❞❛❞❡ ❞❡✿
(a) 10000litros 7% (b) 11000litros 29% (c) 36000litros 53% (d) 24000litros 7% (e) 2400litros 4%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈
❖❜s❡r✈❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ♣❡r❝❡❜❡♠♦s q✉❡ ♠✉✐t♦s ❛❧✉♥♦s ✭✷✾✪✮ ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❇✱ ♦✉ s❡❥❛✱ ❛♣❡♥❛s s♦♠❛r❛♠ ❛s ❞✐♠❡♥sõ❡s ❞♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❡♠ ✈❡③ ❞❡ ♠✉❧t✐♣❧✐❝á✲❧❛s✳ ❆ss✐♠ ❝♦♠♦ ♥❛s q✉❡stõ❡s10 ❡ 11✱ ♠✉✐t♦s ❛❧✉♥♦s ♥ã♦ ❝♦♠♣r❡❡♥❞❡♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ✈♦❧✉♠❡ ❞❡ ✉♠
só❧✐❞♦ s✐♠♣❧❡s ❝♦♠♦ ♦ ❝✉❜♦ ❡ ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ❡ ♣♦r ✐ss♦ ❢❛③❡♠ q✉❛❧q✉❡r ❝á❧❝✉❧♦ q✉❡ t❡♥❤❛ ❛❧❣✉♠❛ ❛❧t❡r♥❛t✐✈❛ ❝♦♠♦ r❡s♣♦st❛✳
◗✉❡stã♦ ✶✸ ✲ ❙❛❜❡♥❞♦ q✉❡ ♦ ✈♦❧✉♠❡ ❞❡ ✉♠ ❝✉❜♦ é64cm3✱ q✉❛❧ ❛ ♠❡❞✐❞❛ ❞❡ s✉❛ ❛r❡st❛❄
(a) 6cm 12% (b) 3cm 9% (c) 4cm 42% (d) 5cm 6% (e) 8cm 31%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❈
◆❡st❛ q✉❡stã♦ q✉❡rí❛♠♦s ✈❡r✐✜❝❛r s❡ ♦ ❛❧✉♥♦ s❛❜✐❛ ❢❛③❡r ♦ ❝á❧❝✉❧♦ ♣❛r❛ ❡♥❝♦♥tr❛r ♦s ✈❛❧♦r❡s ❞♦s ❧❛❞♦s✱ ❞❛❞♦ ♦ ✈❛❧♦r ❞♦ ✈♦❧✉♠❡✳ ❆♣❡♥❛s ✹✷✪ ❛❝❡rt❛r❛♠ ❛ q✉❡stã♦✱ s❡♥❞♦ q✉❡ ✸✶✪ ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❊ q✉❡ t❡♠ ❝♦♠♦ r❡s♣♦st❛ ♦ ♥ú♠❡r♦ q✉❡ ❡❧❡✈❛❞♦ ❛♦ q✉❛❞r❛❞♦ r❡s✉❧t❛ ❡♠ ✻✹✳
◗✉❡stã♦ ✶✹ ✲ ❯♠ ❜❧♦❝♦ r❡t❛♥❣✉❧❛r ❞❡ ♠❛❞❡✐r❛ t❡♠ 160cm ❞❡ ❝♦♠♣r✐♠❡♥t♦✱ 60cm ❞❡ ❧❛r❣✉r❛ ❡ 45cm ❞❡ ❛❧t✉r❛✳ ❖ ❜❧♦❝♦ é ❝♦rt❛❞♦ ✈ár✐❛s ✈❡③❡s✱ ❝♦♠ ❝♦rt❡s ♣❛r❛❧❡❧♦s às s✉❛s ❢❛❝❡s✱ ❞❡ ♠♦❞♦ ❛ s✉❜❞✐✈✐❞✐✲❧♦ ❡♠ ❜❧♦❝♦s t❛♠❜é♠ r❡t❛♥❣✉❧❛r❡s ❞❡ 40cm ❞❡ ❝♦♠♣r✐♠❡♥t♦ ♣♦r20cm ❞❡ ❧❛r❣✉r❛ ♣♦r15cm❞❡ ❛❧t✉r❛✳ ◗✉❛♥t❛s ♣❡ç❛s ❢♦r❛♠ ♦❜t✐❞❛s❄
(a) 36 33% (b) 28 23% (c) 60 24% (d) 40 10% (e) 30 10%
✶✳✷✳ ❆◆➪▲■❙❊ ❉❖❙ ❘❊❙❯▲❚❆❉❖❙
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❆
❯♠❛ ❜♦❛ ♣❛rt❡ ❞♦s ❛❧✉♥♦s ✭✻✼✪✮ ♥ã♦ ❝♦♥s❡❣✉✐✉ ✈✐s✉❛❧✐③❛r ❛ ✐❞❡✐❛ ❞❛ q✉❡stã♦✱ q✉❡ ❡r❛ ♣r❡❡♥❝❤❡r ♦ ♣❛r❛❧❡❧❡♣í♣❡❞♦ ♠❛✐♦r ❝♦♠ ♦s ♣❛r❛❧❡❧❡♣í♣❡❞♦s ♠❡♥♦r❡s✳
◗✉❡stã♦ ✶✺✲ ◗✉❛❧ é ♦ ✈♦❧✉♠❡ ❛♣r♦①✐♠❛❞♦ ❞❡ ✉♠❛ ❡s❢❡r❛ ❞❡ r❛✐♦ 2cm q✉❡ ❡stá ✐♥s❝r✐t❛ ❡♠ ✉♠ ❝✉❜♦ ❝✉❥❛ ❛r❡st❛ ♠❡❞❡ 4cm✱ ❝♦♠♦ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦❄
(a) 64cm3 52% (b) 70cm3 12% (c) 20cm3 11% (d) 34cm3 12% (e) 10cm3 13%
❆❧t❡r♥❛t✐✈❛ ❝♦rr❡t❛✿ ❉
❖ ♦❜❥❡t✐✈♦ ❞❛ ú❧t✐♠❛ q✉❡stã♦ ❡r❛ ✈❡r✐✜❝❛r s❡ ♦ ❛❧✉♥♦ ♠❡s♠♦ ♥ã♦ s❛❜❡♥❞♦ ❛ ❢ór♠✉❧❛ ❞♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛ ❝♦♥s❡❣✉✐r✐❛ ♦❜t❡r ✉♠ ✈❛❧♦r ❛♣r♦①✐♠❛❞♦ ♣❛r❛ s❡✉ ✈♦❧✉♠❡✱ ❛tr❛✈és ❞♦ ✈♦❧✉♠❡ ❞♦ ❝✉❜♦ ❝✐r❝✉♥s❝r✐t♦ à ❡s❢❡r❛✳ ▼❛s ✺✷✪ ♠❛r❝❛r❛♠ ❛ ❧❡tr❛ ❆✱ q✉❡ é ♦ ✈♦❧✉♠❡ ❞♦ ❝✉❜♦ ❡ ♥ã♦ ♦ ✈♦❧✉♠❡ ❞❛ ❡s❢❡r❛✳
✶✳✷ ❆♥á❧✐s❡ ❞♦s r❡s✉❧t❛❞♦s
✶✳✷✳✶ ◗✉❛♥t✐t❛t✐✈♦✴♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s q✉❡ ❛❝❡rt❛r❛♠ ❝❛❞❛ q✉❡s✲
tã♦
❆ t❛❜❡❧❛ ❡ ♦ ❣rá✜❝♦ ❛❜❛✐①♦ ♠♦str❛♠ ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ✭❞❡ ✉♠ t♦t❛❧ ❞❡ ✶✵✼ ♣❛rt✐❝✐♣❛♥t❡s✮ q✉❡ ❛❝❡rt❛r❛♠ ❝❛❞❛ q✉❡stã♦✳ ❖ ♣❡r❝❡♥t✉❛❧ ♠é❞✐♦ ❞❡ ❛❝❡rt♦s ♣♦r q✉❡stã♦ ❢♦✐ ❞❡ ✸✻✪ ❛♣r♦①✐♠❛❞❛♠❡♥t❡✳
◗✉❡stã♦ P❡r❝❡♥t✉❛❧ ❞❡ ❛❝❡rt♦s
✵✶ ✻✷✪
✵✷ ✺✼✪
✵✸ ✷✽✪
✵✹ ✸✺✪
✵✺ ✶✹✪
✵✻ ✻✪
✵✼ ✸✸✪
✵✽ ✶✺✪
✵✾ ✹✼✪
✶✵ ✺✻✪
✶✶ ✹✼✪
✶✷ ✺✸✪
✶✸ ✹✷✪
✶✳✷✳ ❆◆➪▲■❙❊ ❉❖❙ ❘❊❙❯▲❚❆❉❖❙
✶✹ ✸✸✪
✶✺ ✶✷✪
❆♣ós ❝r✐t❡r✐♦s❛ ❛♥á❧✐s❡ ❞❛ t❛❜❡❧❛ ❡ ❞♦ ❣rá✜❝♦✱ ♣♦❞❡✲s❡ ❝♦♥❝❧✉✐r q✉❡ ♦ í♥❞✐❝❡ ❞❡ ❛❝❡rt♦s ♥❛ ♠❛✐♦r✐❛ ❞❛s q✉❡stõ❡s ♥ã♦ ❢♦✐ s❛t✐s❢❛tór✐♦✳ ❆s q✉❡stõ❡s ✺✱ ✻ ❡ ✽ t✐✈❡r❛♠ ✉♠ í♥❞✐❝❡ ❞❡ ❛❝❡rt♦s ♠✉✐t♦ ❜❛✐①♦✳ ❆ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s ❝♦♥❢✉♥❞✐✉ ♦ ❝♦♥❝❡✐t♦ ❞❡ ♣❡rí♠❡tr♦ ❡ ár❡❛✱ ♦✉tr♦s ❛♣❡♥❛s ♠✉❧t✐♣❧✐❝❛r❛♠ ♦s ✈❛❧♦r❡s ❡①♣❧í❝✐t♦s ♥❛s q✉❡stõ❡s ♣♦✐s ♥ã♦ s♦✉❜❡r❛♠ ❞❡❝♦♠♣♦r ❛s ✜❣✉r❛s ❡ ❝❛❧❝✉❧❛r s✉❛s ár❡❛s✳ ❆ q✉❡stã♦ ❝♦♠ ♠❛✐♦r í♥❞✐❝❡ ❞❡ ❛❝❡rt♦s ❢♦✐ ❛ ✶ ❝♦♠ ✻✷✪ ❞❡ ❛❝❡rt♦s✱ ❡♥q✉❛♥t♦ ✐ss♦✱ ❛ q✉❡stã♦ ♦♥❞❡ ♦s ❛❧✉♥♦s ♠❛✐s s❡ ❡q✉✐✈♦❝❛r❛♠ ❢♦✐ ❛ ✻ ❝♦♠ ✻✪ ❞❡ ❛❝❡rt♦s✳ ❆tr❛✈és ❞❛ ❛♥á❧✐s❡ ❞❛ t❛❜❡❧❛ ❡ ❞♦ ❣rá✜❝♦ s❡rá ❢❡✐t♦ ✉♠ ❧❡✈❛♥t❛♠❡♥t♦ ❞♦s ♣♦♥t♦s ♦♥❞❡ ♦s ❡st✉❞❛♥t❡s ❛♣r❡s❡♥t❛♠ ♠❛✐♦r❡s ❞✐✜❝✉❧❞❛❞❡s ❛ r❡s♣❡✐t♦ ❞♦s ❝♦♥❝❡✐t♦s ❛❜♦r❞❛❞♦s✳
✶✳✷✳✷ ◗✉❛♥t✐❞❛❞❡ ❞❡ ❛❝❡rt♦s ❞♦s ❛❧✉♥♦s
❚❛♠❜é♠ ❛♥❛❧✐s❛♠♦s ♦ ♣❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ❡♠ r❡❧❛çã♦ ❛♦ ♥ú♠❡r♦ ❞❡ q✉❡stõ❡s r❡s✲ ♣♦♥❞✐❞❛s ❝♦rr❡t❛♠❡♥t❡ ♣❡❧♦s ♠❡s♠♦s✳ ❆ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s ❛❝❡rt♦✉ ❡♥tr❡ ✹ ❡ ✼ q✉❡stõ❡s ❞❛s ✶✺ ♣r♦♣♦st❛s✱ ✉♠ ❛❧✉♥♦ ♥ã♦ ❛❝❡rt♦✉ ♥❡♥❤✉♠❛ q✉❡stã♦✱ ❡ s♦♠❡♥t❡ ✉♠ ❛❝❡rt♦✉ ✶✺ q✉❡s✲ tõ❡s✱ ♦ q✉❡ é ✉♠ ♥ú♠❡r♦ ❜❛✐①♦✱ ❧❡✈❛♥❞♦ ❡♠ ❝♦♥s✐❞❡r❛çã♦ ♦ ♥í✈❡❧ ❞♦ t❡st❡ ❡ ❛ sér✐❡ ❞❡❧❡s✳ ❆ t❛❜❡❧❛ ❛❜❛✐①♦ ♠♦str❛ ❡ss❡s ❞❛❞♦s✳
P❡r❝❡♥t✉❛❧ ❞❡ ❛❧✉♥♦s ◆ú♠❡r♦ ❞❡ ❛❝❡rt♦s
✶✪ ✵✵
✼✪ ✵✶
✽✪ ✵✷
✽✪ ✵✸
✶✳✷✳ ❆◆➪▲■❙❊ ❉❖❙ ❘❊❙❯▲❚❆❉❖❙
✶✻✪ ✵✹
✶✹✪ ✵✺
✶✸✪ ✵✻
✶✸✪ ✵✼
✾✪ ✵✽
✹✪ ✵✾
✷✪ ✶✵
✷✪ ✶✶
✶✪ ✶✷
✶✪ ✶✸
✵✪ ✶✹
✶✪ ✶✺
❆♥❛❧✐s❛♥❞♦ ♦s r❡s✉❧t❛❞♦s ♦❜t✐❞♦s ♣❡r❝❡❜❡♠♦s q✉❡ ❛ ♠❛✐♦r✐❛ ❞♦s ❛❧✉♥♦s✱ ❡♠❜♦r❛ ❥á ❡st❡❥❛♠ ♥♦2♦ ❛♥♦ ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ❛✐♥❞❛ ♥ã♦ ❝♦♠♣r❡❡♥❞❡♠ ♦s ❝♦♥❝❡✐t♦s ❞❡ ár❡❛ ❞❡ ✜❣✉r❛s
♣❧❛♥❛s ❡ ✈♦❧✉♠❡ ❞❡ só❧✐❞♦s ❣❡♦♠étr✐❝♦s✳ ▼✉✐t♦s ❝♦♥❢✉♥❞❡♠ ár❡❛ ❝♦♠ ♣❡rí♠❡tr♦✱ ✈♦❧✉♠❡ ❝♦♠ ár❡❛ ♦✉ ❛♣❡♥❛s ❢❛③❡♠ ❝á❧❝✉❧♦s s❡♠ s❡♥t✐❞♦ ♣❛r❛ ❛ q✉❡stã♦✳ ❈♦♠ ❜❛s❡ ♥❡ss❡s ❞❛❞♦s ❢♦✐ ♣r♦❞✉③✐❞♦ ♦ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ ♣❛r❛ ❛s ❛✉❧❛s✱ q✉❡ s❡rá ♦ ❝❛♣ít✉❧♦ ■■ ❞❡ss❡ tr❛❜❛❧❤♦✳ ❈♦♠ ♦ ♠❛t❡r✐❛❧ ♣r♦❞✉③✐❞♦ ♣r❡t❡♥❞❡♠♦s✿
• ❚r❛❜❛❧❤❛r ♦s ❝♦♥❝❡✐t♦s ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡✱ ♣♦✐s ♠✉✐t♦s ❛❧✉♥♦s ♦s ❝♦♥❢✉♥❞❡♠ ❝♦♠
♣❡rí♠❡tr♦✳
• ❉❡❞✉③✐r ♦✉ ❞❡♠♦♥str❛r ❛s ❢ór♠✉❧❛s ♣❛r❛ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡s ❞❡ ♠❛♥❡✐r❛
q✉❡ ♦s ❛❧✉♥♦s ❝♦♠♣r❡❡♥❞❛♠ ❛s ♠❡s♠❛s✱ ♣❛r❛ q✉❡ ♥ã♦ s♦♠❡♥t❡ ❢❛ç❛♠ ❝á❧❝✉❧♦s s❡♠ s✐❣♥✐✜❝❛❞♦✳
• ❉❡❝♦♠♣♦r ✉♠❛ ✜❣✉r❛ ❡♠ ♦✉tr❛s ♠❛✐s s✐♠♣❧❡s ❡ ❝❛❧❝✉❧❛r s✉❛ ár❡❛✳
• ❈❛❧❝✉❧❛r ár❡❛ ❡ ✈♦❧✉♠❡ ♣♦r ❛♣r♦①✐♠❛çã♦✳
• ❈❛❧❝✉❧❛r ❛ ár❡❛ ❞❛ ♣❛rt❡ ❤❛❝❤✉r❛❞❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ♦✉ ✈♦❧✉♠❡ q✉❡ s♦❜r❛ ♦✉ q✉❡
❢❛❧t❛ ❞❡♣♦✐s ❞❛ ✐♥s❝r✐çã♦ ❞❡ ✉♠ só❧✐❞♦ ❡♠ ♦✉tr♦✳
• ❈♦♥❤❡❝❡r ❛ r❡❧❛çã♦ ❡♥tr❡ ✈♦❧✉♠❡ ❡ ❝❛♣❛❝✐❞❛❞❡ ❞❡ ✉♠ só❧✐❞♦✳
❈❛♣ít✉❧♦ ✷
❊st✉❞♦ ❞❡ ár❡❛s ❡ ✈♦❧✉♠❡
◆❡st❡ ❝❛♣ít✉❧♦ ❛♣r❡s❡♥t❛r❡♠♦s ♦ ♠❛t❡r✐❛❧ ❞❡ ❛♣♦✐♦ q✉❡ ❢♦✐ ✉s❛❞♦ ♥❛ s❛❧❛ ❞❡ ❛✉❧❛ ♥❛s t✉r♠❛s q✉❡ ✜③❡r❛♠ ♦ t❡st❡ ❞✐❛❣♥óst✐❝♦ s♦❜r❡ ♦s ❝♦♥❝❡✐t♦s ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡ ❛♣r❡s❡♥t❛❞♦ ♥♦ ❝❛♣ít✉❧♦ ✶✳ ❋♦❝❛♠♦s ♦ ❡st✉❞♦ ❞❡ ár❡❛s ❞❡ ✜❣✉r❛s ♣❧❛♥❛s✱ q✉❛♥t♦ ❛♦ ✈♦❧✉♠❡ ❡①♣❧♦r❛✲ r❡♠♦s ❛♣❡♥❛s ♦ ❞❡ ❛❧❣✉♥s só❧✐❞♦s✳ Pr✐♠❡✐r♦ ❛♣r❡s❡♥t❛♠♦s ♦s ❝♦♥❝❡✐t♦s ❞❡ ár❡❛ ❡ ✈♦❧✉♠❡✱ ❞❡❞✉③✐♠♦s ❛❧❣✉♠❛s ❢ór♠✉❧❛s ❡ ❞❡♣♦✐s ❞❡s❡♥✈♦❧✈❡♠♦s ❛❧❣✉♠❛s ❛t✐✈✐❞❛❞❡s ❝♦♠ ♠❛t❡r✐❛✐s ❞♦ ▲❛❜♦r❛tór✐♦ ❞❡ ▼❛t❡♠át✐❝❛ q✉❡ ❢❛❝✐❧✐t❛♠ ❛ ❛♣r❡♥❞✐③❛❣❡♠ ❞♦s ❝♦♥t❡ú❞♦s✳
❖ t❡①t♦ ❛ s❡❣✉✐r ❢♦✐ ❜❛s❡❛❞♦ ♥♦s ❧✐✈r♦s ❞♦ ❡♥s✐♥♦ ♠é❞✐♦ ✿ ❈♦♥t❡①t♦ ❡ ❛♣❧✐❝❛çõ❡s ❞❡ ▲✉✐③ ❘♦❜❡rt♦ ❉❛♥t❡✱ ❈✐ê♥❝✐❛ ❡ ❛♣❧✐❝❛çõ❡s ❞❡ ●❡❧s♦♥ ■❡③③✐✱ ▼❛t❡♠át✐❝❛ ❞❡ ▼❛♥♦❡❧ P❛✐✈❛✱ ◆♦✈♦ ❖❧❤❛r ▼❛t❡♠át✐❝❛ ❞❡ ❏♦❛♠✐r ❙♦✉③❛ ❡ ♥♦s ❧✐✈r♦s ❞❛ ❙❇▼✿ ❚ó♣✐❝♦s ❞❡ ▼❛t❡♠át✐❝❛ ❊❧❡✲ ♠❡♥t❛r ❞❡ ❆♥tô♥✐♦ ❈❛♠✐♥❤❛ ▼✉♥✐③ ◆❡t♦ ❡ ❆ ▼❛t❡♠át✐❝❛ ❞♦ ❊♥s✐♥♦ ▼é❞✐♦ ❞❡ ❊❧♦♥ ▲❛❣❡s✳
✷✳✶ ➪r❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s
❖ ❡st✉❞♦ ❞❛ ár❡❛ ❞❡ ✜❣✉r❛s ♣❧❛♥❛s ❡stá ❧✐❣❛❞♦ ❛♦s ❝♦♥❝❡✐t♦s r❡❧❛❝✐♦♥❛❞♦s à ●❡♦♠❡tr✐❛ ❊✉❝❧✐❞✐❛♥❛✱ q✉❡ s✉r❣✐✉ ♥❛ ●ré❝✐❛ ❛♥t✐❣❛ ❡♠❜❛s❛❞❛ ♥♦ ❡st✉❞♦ ❞♦ ♣♦♥t♦✱ ❞❛ r❡t❛ ❡ ❞♦ ♣❧❛♥♦✳ ❖s ❡❣í♣❝✐♦s t❛♠❜é♠ ✉t✐❧✐③❛✈❛♠ ♦ ❝♦♥❝❡✐t♦ ❞❡ ár❡❛s ❤á ♠✐❧❤❛r❡s ❞❡ ❛♥♦s✳ ❉❡♣♦✐s ❞❛s ❝❤❡✐❛s ❞♦ r✐♦ ◆✐❧♦✱ ❛s á❣✉❛s ❝♦♠❡ç❛✈❛♠ ❛ ❜❛✐①❛r ❡ ❛s ♠❛r❣❡♥s ✜❝❛✈❛♠ ❝♦❜❡rt❛s ♣♦r ✉♠❛ ❧❛♠❛ ❝♦♥t❡♥❞♦ ✈ár✐♦s ♥✉tr✐❡♥t❡s✱ q✉❡ t♦r♥❛✈❛♠ ♦ s♦❧♦ ❢ért✐❧ ♣❛r❛ ❛ ❛❣r✐❝✉❧t✉r❛✳ ◆♦ ❡♥t❛♥t♦✱ ❛♦ ❜❛✐①❛r❡♠ ❛s á❣✉❛s✱ ❛s ❞❡♠❛r❝❛çõ❡s q✉❡ ❞❡❧✐♠✐t❛✈❛♠ ❛s ♣r♦♣r✐❡❞❛❞❡s ❡r❛♠ ❞❡s❢❡✐t❛s✱ s❡♥❞♦ ♥❡❝❡ssár✐❛ ❛ r❡❛❧✐③❛çã♦ ❞❡ ♥♦✈❛s ♠❡❞✐çõ❡s✳ ▼✉✐t♦s r❡❣✐str♦s ❡♥✈♦❧✈❡♥❞♦ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s ♣♦❞❡♠ s❡r ❡♥❝♦♥tr❛❞♦s ♥♦ ♣❛♣✐r♦ ❞❡ ❘❤✐♥❞✱ ✐♠♣♦rt❛♥t❡ ❞♦❝✉♠❡♥t♦ ❡❣í♣❝✐♦ sé❝✉❧♦ ❳❱■■ ❛✳❈✳
◆♦ ♠✉♥❞♦ ❡♠ q✉❡ ✈✐✈❡♠♦s✱ ♦ ❝á❧❝✉❧♦ ❞❡ ár❡❛s t❡♠ ♠✉✐t❛ ❛♣❧✐❝❛❜✐❧✐❞❛❞❡ ❡♠ ❞✐❢❡r❡♥t❡s ♠♦♠❡♥t♦s✱ s❡❥❛ ❡♠ ❛t✐✈✐❞❛❞❡s ♣✉r❛♠❡♥t❡ ❝♦❣♥✐t✐✈❛s✱ ♦✉ ❛té ♠❡s♠♦ tr❛❜❛❧❤✐st❛s✳ ❯♠ ❡①❡♠♣❧♦ ❞❡ ♣r♦✜ss✐♦♥❛❧ q✉❡ ❢❛③ ✉s♦ ❞❡ss❛ ❢❡rr❛♠❡♥t❛ ♣❛r❛ t♦r♥❛r ♣♦ssí✈❡❧ ♦ ❞❡s❡♠♣❡♥❤♦ ❞♦ s❡✉ tr❛❜❛❧❤♦ é ♦ ❊♥❣❡♥❤❡✐r♦ ❈✐✈✐❧✳ ➱ ❛tr❛✈és ❞♦ ❝♦♥❤❡❝✐♠❡♥t♦ ❞❡ ár❡❛ q✉❡ é ♣♦ssí✈❡❧ ❡st✐♠❛r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ❝❡râ♠✐❝❛ ♥❡❝❡ssár✐❛ ♣❛r❛ ♣❛✈✐♠❡♥t❛r ✉♠ ❞❡t❡r♠✐♥❛❞♦ ❝ô♠♦❞♦ ❞❡ ✉♠❛ ❝❛s❛✱ ♣♦r ❡①❡♠♣❧♦✳
✷✳✶✳ ➪❘❊❆ ❉❊ ❋■●❯❘❆❙ P▲❆◆❆❙
✷✳✶✳✶ ❈♦♥❝❡✐t♦ ❞❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛
❆ ✐❞❡✐❛ ❞❡ ár❡❛ ❡stá r❡❧❛❝✐♦♥❛❞❛ à ♠❡❞✐❞❛ ❞❡ ✉♠❛ s✉♣❡r❢í❝✐❡✳ ❆ ár❡❛ ❞❡ ✉♠❛ r❡❣✐ã♦ ♦✉ s✉♣❡r❢í❝✐❡ ♣♦❞❡ s❡r ♦❜t✐❞❛ r❡❧❛❝✐♦♥❛♥❞♦✲❛ ❝♦♠ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✳ ❙❡ q✉✐s❡r♠♦s ♠❡❞✐r ❛ ár❡❛ ❞❛ r❡❣✐ã♦ ❤❛❝❤✉r❛❞❛ ❞❛ ✜❣✉r❛ ✭✷✳✶✮✱ ♣r❡❝✐s❛♠♦s ❝♦♠♣❛rá✲❧❛ ❝♦♠ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛U✳ ❖ r❡s✉❧t❛❞♦ ❞❡ss❛ ❝♦♠♣❛r❛çã♦ é ✉♠ ♥ú♠❡r♦ q✉❡ ❡①♣r✐♠❡ q✉❛♥t❛s ✈❡③❡s ❛ ✉♥✐❞❛❞❡ ❞❡ ár❡❛U ❝❛❜❡ ♥❛ r❡❣✐ã♦ ❞❡st❛❝❛❞❛✳
❋✐❣✉r❛ ✷✳✶✿ ❯♥✐❞❛❞❡ ❞❡ ár❡❛
❖❜s❡r✈❛♥❞♦ ❛ ♣❛rt❡ ❞❡st❛❝❛❞❛ ❞❛ ✜❣✉r❛ ✷✳✶✱ ♣♦❞❡♠♦s ♥♦t❛r q✉❡ sã♦ ♥❡❝❡ssár✐♦s28,5U ♣❛r❛ ❝♦❜r✐✲❧❛✳ ❉✐③❡♠♦s ❡♥tã♦ q✉❡ ❛ ár❡❛ ❞❛ ✜❣✉r❛ é28,5U✳
❙❡ ❝♦♥s✐❞❡ráss❡♠♦s q✉❛❧q✉❡r ❝♦✐s❛ ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ár❡❛✱ ✉♠❛ ♠❡s♠❛ ✜❣✉r❛ t❡r✐❛ ✈ár✐♦s ✈❛❧♦r❡s ❝♦♠♦ ár❡❛✳ P❛r❛ q✉❡ ✐st♦ ♥ã♦ ❛❝♦♥t❡ç❛ ❡st❛❜❡❧❡❝❡r❡♠♦s ❝♦♠♦ ✉♥✐❞❛❞❡ ❞❡ ár❡❛ ✉♠❛ r❡❣✐ã♦ q✉❛❞r❛❞❛ ❝✉❥♦ ❧❛❞♦ ♠❡❞❡ ✉♠❛ ✉♥✐❞❛❞❡ ❞❡ ❝♦♠♣r✐♠❡♥t♦✳ ❆ ✉♥✐❞❛❞❡ ♣❛❞rã♦ ❞❡ ár❡❛ é ♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ 1✳ P♦r ❡①❡♠♣❧♦✱ s❡ ♦ q✉❛❞r❛❞♦ t❡♠ ❧❛❞♦s ❞❡ 1m ❛ s✉❛ ár❡❛ é ✉♠ ♠❡tr♦ q✉❛❞r❛❞♦m2✳
❆ s❡❣✉✐r ✈❛♠♦s ❡st✉❞❛r ♦ ❝á❧❝✉❧♦ ❞❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❡ ❛❧❣✉♠❛s ✜❣✉r❛s ♣❧❛♥❛s ❡ s✉❛s r❡s♣❡❝t✐✈❛s ❢ór♠✉❧❛s✳
✷✳✶✳✷ ➪r❡❛ ❞♦ q✉❛❞r❛❞♦
❈♦♥s✐❞❡r❡♠♦s ✉♠❛ r❡❣✐ã♦ q✉❛❞r❛❞❛ ◗ ❝✉❥♦ ❧❛❞♦ ♠❡❞❡n✱ ♦♥❞❡n é ✉♠ ♥ú♠❡r♦ ♥❛t✉r❛❧✳ ❊❧❛ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st❛ ❡♠n2r❡❣✐õ❡s q✉❛❞r❛❞❛s ❥✉st❛♣♦st❛s✱ ❝❛❞❛ ✉♠❛ ❝♦♠ ❧❛❞♦ ✉♥✐tár✐♦
❡✱ ♣♦rt❛♥t♦✱ ❝♦♠ ár❡❛1✳ ▲♦❣♦✱ ❛ r❡❣✐ã♦ q✉❛❞r❛❞❛ t❡♠ ár❡❛ n2✳
✷✳✶✳ ➪❘❊❆ ❉❊ ❋■●❯❘❆❙ P▲❆◆❆❙
❘❡❣✐ã♦ q✉❛❞r❛❞❛ ❞❡ ❧❛❞♦ 3✱ ❞❡❝♦♠♣♦st❛ ❡♠ 9 = 32 r❡❣✐õ❡s q✉❛❞r❛❞❛s ✉♥✐tár✐❛s✳ ❆
♠❡s♠❛ ✐❞❡✐❛ ✈❛❧❡ ♣❛r❛ q✉❛❧q✉❡r ♥ú♠❡r♦ r❡❛❧ ♣♦s✐t✐✈♦✳ ❈♦♠ ✐ss♦ ❞❡❞✉③✐♠♦s q✉❡ ❛ ♠❡❞✐❞❛ ❞❛ ár❡❛ ❞❡ q✉❛❧q✉❡r q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦ l é ❞❛❞♦ ♣♦r✿
➪r❡❛ ❞♦ q✉❛❞r❛❞♦ =l2.
❆ ár❡❛ ❞♦ q✉❛❞r❛❞♦ é ✐❣✉❛❧ ❛♦ q✉❛❞r❛❞♦ ❞❛ ♠❡❞✐❞❛ ❞❡ s❡✉ ❧❛❞♦✳
❊①❡♠♣❧♦✿ P❛r❛ ❝♦❧♦❝❛r ❝❡râ♠✐❝❛ ♥❛ s❛❧❛ ❞❡ s✉❛ ❝❛s❛✱ ❉✳ ▲✉③✐❛ ❝♦♠♣r♦✉ 26m2 ❞❡ ♣✐s♦✳
❙❛❜❡♥❞♦ q✉❡ ❛ s❛❧❛ t❡♠ ♦ ❢♦r♠❛t♦ q✉❛❞r❛♥❣✉❧❛r ❡ q✉❡ ✉♠ ❞♦s ❧❛❞♦s ♠❡❞❡5m✱ ❞✐❣❛ s❡ ♦ ♣✐s♦ ❝♦♠♣r❛❞♦ ♣♦r ❉✳ ▲✉③✐❛ s❡rá s✉✜❝✐❡♥t❡ ♣❛r❛ ♣❛✈✐♠❡♥t❛r ❛ s✉❛ s❛❧❛✳
❘❡s♦❧✉çã♦✿ ❆ s❛❧❛ t❡♠ ♦ ❢♦r♠❛t♦ q✉❛❞r❛♥❣✉❧❛r ❡ s❡✉ ❧❛❞♦ ♠❡❞❡5m✳ ➪r❡❛= 52 = 25m2.
❈♦♠♦ ❡❧❛ ❝♦♠♣r♦✉26m2 ❡ ❛ ár❡❛ ❞❛ s❛❧❛ só é ❞❡25m2✱ ❡♥tã♦ ❛ ❝❡râ♠✐❝❛ s❡rá ♦ s✉✜❝✐❡♥t❡
♣❛r❛ ♣❛✈✐♠❡♥t❛r ❛ s✉❛ s❛❧❛✳
✷✳✶✳✸ ➪r❡❛ ❞♦ r❡tâ♥❣✉❧♦
❆ r❡❣✐ã♦ r❡t❛♥❣✉❧❛r ❛❜❛✐①♦ ♣✐♥t❛❞❛ ❝♦♥té♠ ✶✷ ✉♥✐❞❛❞❡s ❞❡ ár❡❛✳ P♦rt❛♥t♦✱ s✉❛ ár❡❛ é ❞❡12cm2✳
❖❜s❡r✈❡ q✉❡✱ ❡♠ ✈❡③ ❞❡ ❝♦♥t❛r q✉❛♥t❛s ✉♥✐❞❛❞❡s ❞❡ ár❡❛ ❡stã♦ ❝♦♥t✐❞❛s ♥❛ r❡❣✐ã♦ r❡t❛♥❣✉❧❛r✱ ❜❛st❛ ♠✉❧t✐♣❧✐❝❛r ❛ ♠❡❞✐❞❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ♣❡❧❛ ♠❡❞✐❞❛ ❞❛ ❧❛r❣✉r❛✳
3cm·4cm= 12cm2.
◆♦ ❡①❡♠♣❧♦ ♠♦str❛❞♦ ❛❝✐♠❛ ✉s❛♠♦s ♥ú♠❡r♦s ♥❛t✉r❛✐s✳ ❱❛♠♦s ♣r♦✈❛r q✉❡✱ s❡ ❛ ♠❡❞✐❞❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦c❡ ❛ ♠❡❞✐❞❛ ❞❛ ❧❛r❣✉r❛l ❢♦r❡♠ ♥ú♠❡r♦s r❡❛✐s q✉❛✐sq✉❡r✱ ❛ ár❡❛ ❞❛ r❡❣✐ã♦ r❡t❛♥❣✉❧❛rR é ❞❛❞❛ ♣♦r✿
➪r❡❛ ❞♦ r❡tâ♥❣✉❧♦ ❂ c·l.
❈♦♥s✐❞❡r❡♠♦s ✉♠❛ r❡❣✐ã♦ r❡t❛♥❣✉❧❛rR ❞❡ ❝♦♠♣r✐♠❡♥t♦c❡ ❧❛r❣✉r❛l✱ ❡♠ q✉❡c❡ lsã♦ ♥ú♠❡r♦s r❡❛✐s✳
✷✳✶✳ ➪❘❊❆ ❉❊ ❋■●❯❘❆❙ P▲❆◆❆❙
❈♦♥str✉í♠♦s ✉♠❛ r❡❣✐ã♦ q✉❛❞r❛❞❛ ❝✉❥❛ ♠❡❞✐❞❛ ❞♦ ❧❛❞♦c+l✱ q✉❡ ❝♦♥té♠ ❞♦✐s r❡tâ♥✲
❣✉❧♦s R ❡ ❞✉❛s r❡❣✐õ❡s q✉❛❞r❛❞❛s✱ ✉♠❛ ❞❡ ❧❛❞♦ c❡ ♦✉tr❛ ❞❡ ❧❛❞♦ l✳
❆ ár❡❛ ❞❡ss❛ r❡❣✐ã♦ q✉❛❞r❛❞❛ ❞❡ ❧❛❞♦ c+l é ❞❛❞❛ ♣❡❧♦ q✉❛❞r❛❞♦ ❞❛ s♦♠❛✿
➪r❡❛ ❞♦ q✉❛❞r❛❞♦= (c+l)2 =c2+ 2cl+l2. ✭✷✳✶✮ ❈♦♠♦ ❛s r❡❣✐õ❡s q✉❛❞r❛❞❛s ❞❡ ❧❛❞♦s c ❡ l tê♠ ár❡❛s✱ r❡s♣❡❝t✐✈❛♠❡♥t❡ ✐❣✉❛✐s ❛ l2 ❡c2✱
❝♦♥❝❧✉í♠♦s q✉❡ ❛ ár❡❛ ❞♦ q✉❛❞r❛❞♦ ❞❡ ❧❛❞♦c+l é✿
➪r❡❛ ❞♦ q✉❛❞r❛❞♦=c2+l2+ 2·✭ár❡❛ ❞❡ ❘✮. ✭✷✳✷✮ ❈♦♠♣❛r❛♥❞♦ ❛s ❡①♣r❡ssõ❡s (2.1) ❡ (2.2)✱ t❡♠♦s✿
c2+ 2cl+l2 =c2+l2+ 2
·✭ár❡❛ ❞❡ ❘✮ =⇒ ✷❝❧=✷·✭ár❡❛ ❞❡ ❘✮
➪r❡❛ ❞♦ ❘❡tâ♥❣✉❧♦ ❘ ❂ ❝·l.
❆ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛ ♠❡❞✐❞❛ ❞♦ ❝♦♠♣r✐♠❡♥t♦ ♣❡❧❛ ♠❡❞✐❞❛ ❞❛ ❧❛r❣✉r❛✳
❊①❡♠♣❧♦✿ ❯♠ ❞♦s ❧❛❞♦s ❞♦ r❡tâ♥❣✉❧♦ 1 ♠❡❞❡ 18cm✳ ◗✉❛❧ ❞❡✈❡ s❡r ❛ ♠❡❞✐❞❛ ❞♦ ♦✉tr♦ ❧❛❞♦ ♣❡r♣❡♥❞✐❝✉❧❛r ♣❛r❛ q✉❡ ❛ ár❡❛ ❞♦ r❡tâ♥❣✉❧♦1 s❡❥❛ ❡q✉✐✈❛❧❡♥t❡ à ár❡❛ ❞♦ r❡tâ♥❣✉❧♦2
❝✉❥♦s ❧❛❞♦s ♠❡❞❡♠9cm ❡ 12cm❄
❘❡s♦❧✉çã♦✿ ❈♦♠♦ ❥á ❝♦♥❤❡❝❡♠♦s ❛s ♠❡❞✐❞❛s ❞❡ ❝♦♠♣r✐♠❡♥t♦ ❡ ❧❛r❣✉r❛ ❞♦ r❡tâ♥❣✉❧♦
2✱ ♣♦❞❡♠♦s ❝❛❧❝✉❧❛r s✉❛ ár❡❛ q✉❡ é✿ A = 12 ·9 = 108cm2✳ ❖ r❡tâ♥❣✉❧♦ 1 t❡♠ ❛ ár❡❛
❡q✉✐✈❛❧❡♥t❡ à ❝❛❧❝✉❧❛❞❛ ❛❝✐♠❛✱ s❛❜❡♠♦s q✉❡ ✉♠ ❞❡ s❡✉s ❧❛❞♦s t❡♠ 18cm✱ q✉❡r❡♠♦s s❛❜❡r q✉❛♥t♦ ♠❡❞❡ s❡✉ ♦✉tr♦ ❧❛❞♦✳ P❛r❛ ✐ss♦ ❜❛st❛ r❡s♦❧✈❡r ❛ ❡①♣r❡ssã♦ ❛❜❛✐①♦ ❡ ❡♥❝♦♥tr❛r ❛ ♠❡❞✐❞❛ ❞♦ ♦✉tr♦ ❧❛❞♦✿
108 = 18·❧=⇒l= 108
18 = 6cm.
▲♦❣♦✱ ♦ ♦✉tr♦ ❧❛❞♦ ❞♦ r❡tâ♥❣✉❧♦ 1 ♠❡❞❡ 6cm✳
✷✳✶✳ ➪❘❊❆ ❉❊ ❋■●❯❘❆❙ P▲❆◆❆❙
✷✳✶✳✹ ➪r❡❛ ❞♦ P❛r❛❧❡❧♦❣r❛♠♦
❉❡t❡r♠✐♥❡♠♦s ❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABCD t♦♠❛♥❞♦ ❝♦♠♦ ❜❛s❡ ♦ ❧❛❞♦ AB ❞❡ ♠❡❞✐❞❛b ❡ ❛❧t✉r❛ CE ✭♣❡r♣❡♥❞✐❝✉❧❛r ❛ AB✮ ❞❡ ♠❡❞✐❞❛ h.
❆ r❡❣✐ã♦ ❧✐♠✐t❛❞❛ ♣❡❧♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ❡stá ❝♦♥t✐❞❛ ❡♠ ✉♠❛ r❡❣✐ã♦ r❡t❛♥❣✉❧❛r ❞❡ ❜❛s❡ b+c❡ ❛❧t✉r❛ h✳ ❆ ár❡❛ ❞❡ss❡ r❡tâ♥❣✉❧♦ é ❞❛❞❛ ♣♦r✿
(b+c)h=bh+ch.
▼❛s ♦ r❡tâ♥❣✉❧♦AECF é ❢♦r♠❛❞♦ ♣❡❧♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ♠❛✐s ❞♦✐s tr✐â♥❣✉❧♦s ❝♦♥❣r✉❡♥t❡s ❞❡ ❜❛s❡ ❝ ❡ ❛❧t✉r❛ ❤✳ ❖s ❞♦✐s tr✐â♥❣✉❧♦s ❥✉♥t♦s ❢♦r♠❛♠ ♦ r❡tâ♥❣✉❧♦ ❞❡ ár❡❛ c·h✳ ▲♦❣♦ r❡t✐r❛♥❞♦✲s❡ ❡ss❡ r❡tâ♥❣✉❧♦ ❞❡ ár❡❛c·h✱ ❞♦ r❡tâ♥❣✉❧♦AECF ❞❡ ár❡❛ (b+c)h t❡♠♦s q✉❡✿
➪r❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦=b·h.
❆ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞♦ ❧❛❞♦ ♣❡❧❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛ ❡ss❡ ❧❛❞♦✳
❊①❡♠♣❧♦✿ ❈❛❧❝✉❧❡ ❛ ár❡❛ ❞❛ ✜❣✉r❛ ❛❜❛✐①♦✿
❘❡s♦❧✉çã♦✿ P♦❞❡♠♦s ♦❜s❡r✈❛r q✉❡ ❛ ✜❣✉r❛ ❛❝✐♠❛ ♣♦❞❡ s❡r ❞❡❝♦♠♣♦st❛ ❡♠ ❞♦✐s ♣❛r❛❧❡❧♦✲ ❣r❛♠♦s ❞❡ ❜❛s❡ 20cm ❡ ❛❧t✉r❛ ❞❡ 15cm✳
❊♥tã♦ ❛ ár❡❛ ❞❛ ✜❣✉r❛ é ❛ s❡❣✉✐♥t❡✿ A= 20·15·2 = 600cm2.
✷✳✶✳ ➪❘❊❆ ❉❊ ❋■●❯❘❆❙ P▲❆◆❆❙
✷✳✶✳✺ ➪r❡❛ ❞♦ tr✐â♥❣✉❧♦
❙❡❥❛ ♦ tr✐â♥❣✉❧♦ ABC✱ ❝✉❥❛ ❜❛s❡ AC ♠❡❞❡ b ❡ ❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛ ❡ss❛ ❜❛s❡ ♠❡❞❡ h✱ r❡♣r❡s❡♥t❛❞♦ ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✿
◆♦t❡ q✉❡ ❛s r❡s♣❡❝t✐✈❛s ♣❛r❛❧❡❧❛s ❛♦s ❧❛❞♦s AC ❡ AB✱ tr❛ç❛❞❛s ♣❡❧♦s ✈ért✐❝❡s B ❡ C✱ ✐♥t❡r❝❡♣t❛♠✲s❡ ♥♦ ♣♦♥t♦ D✱ ❞❡t❡r♠✐♥❛♥❞♦ ❛ss✐♠ ♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABDC✱ ❝✉❥❛s ♠❡❞✐❞❛s ❞❛ ❜❛s❡ ❡ ❞❛ ❛❧t✉r❛ sã♦b ❡ h✱ ❝♦♥❢♦r♠❡ ♠♦str❛ ❛ ✜❣✉r❛ ❛❜❛✐①♦✳
❈♦♠♦ ❛ med(AB) = med(DC)✱ med(BACˆ ) = med(BDCˆ ) ❡ med(AC) = med(BD)✱
♦s tr✐â♥❣✉❧♦s ABC ❡ DCB sã♦ ❝♦♥❣r✉❡♥t❡s ✭♣❡❧♦ ❝❛s♦ ❞❡ ❝♦♥❣r✉ê♥❝✐❛ ▲❆▲✿ tê♠ ✉♠ â♥❣✉❧♦ ❝♦♠✉♠ ❝♦♠♣r❡❡♥❞✐❞♦ ❡♥tr❡ ❞♦✐s ❧❛❞♦s ❞❡ ♠❡s♠❛ ♠❡❞✐❞❛✮✱ ❧♦❣♦ s✉❛s ár❡❛s sã♦ ✐❣✉❛✐s✳ P♦rt❛♥t♦✱ ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦ ABC é ✐❣✉❛❧ à ♠❡t❛❞❡ ❞❛ ár❡❛ ❞♦ ♣❛r❛❧❡❧♦❣r❛♠♦ ABDC✳
➪r❡❛ ❞♦ tr✐â♥❣✉❧♦= b·h 2 .
❆ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ é ✐❣✉❛❧ à ♠❡t❛❞❡ ❞♦ ♣r♦❞✉t♦ ❞❛ ♠❡❞✐❞❛ ❞❛ ❜❛s❡ ♣❡❧❛ ♠❡❞✐❞❛ ❞❛ ❛❧t✉r❛ r❡❧❛t✐✈❛ ❛ ❡ss❛ ❜❛s❡✳
➪r❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦
❙❡ ♦ tr✐â♥❣✉❧♦ ❢♦r ❡q✉✐❧át❡r♦ t♦❞♦s ♦s ❧❛❞♦s ❡ â♥❣✉❧♦s sã♦ ❝♦♥❣r✉❡♥t❡s ❡ t♦❞❛ ❛❧t✉r❛ é t❛♠❜é♠ ♠❡❞✐❛♥❛ ❡ ❜✐ss❡tr✐③✳ P♦❞❡♠♦s ❞❡t❡r♠✐♥❛r ❛ ❛❧t✉r❛ h ❡♠ ❢✉♥çã♦ ❞❡ l✱ ✉s❛♥❞♦ ♦ t❡♦r❡♠❛ ❞❡ P✐tá❣♦r❛s ❝♦♠♦ ♠♦str❛♠♦s ❛ s❡❣✉✐r✿
✷✳✶✳ ➪❘❊❆ ❉❊ ❋■●❯❘❆❙ P▲❆◆❆❙
l2 =h2 +
l
2
2
h2 =l2−
l
2
2
h2 = 3l
2
4
h= l
√
3 2 .
❊♥tã♦ ❛ ár❡❛ ❞❡❧✐♠✐t❛❞❛ ♣❡❧♦ tr✐â♥❣✉❧♦ ❆❇❈ é ❞❛❞❛ ♣♦r✿
A= base·altura 2 =
l· l√3 2
2 =l·
l√3 2 ·
1 2 =
l2√3
4 .
P♦rt❛♥t♦✱ ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦ ❡q✉✐❧át❡r♦ ❞❡ ❧❛❞♦ ❧ é ❞❛❞❛ ♣♦r✿
A= l
2√3
4 .
❊①❡♠♣❧♦✿ ◗✉❛♥t♦s cm2 ❞❡ ♣❛♣❡❧ sã♦ ♥❡❝❡ssár✐♦s ♣❛r❛ ❢❛③❡r ❛s q✉❛tr♦ ❜❛♥❞❡✐r✐♥❤❛s
❛❜❛✐①♦✱ s❛❜❡♥❞♦ q✉❡ t♦❞❛s sã♦ ✐❣✉❛✐s❄
❘❡s♦❧✉çã♦✿ ❈♦♠♦ ♦s tr✐â♥❣✉❧♦s sã♦ ❡q✉✐❧át❡r♦s ❞❡ ❧❛❞♦ l = 20cm✱ ♣♦❞❡♠♦s ✉s❛r ❛ ❢ór✲ ♠✉❧❛✿ A=l2√3/4✱ ❡ ♠✉❧t✐♣❧✐❝❛r ♣♦r 4✱ ♣♦✐s t♦❞♦s sã♦ ✐❣✉❛✐s✳
A= 20
2√3
4 ·4 = 20
2√3 = 400√3cm2.
❙❡ q✉✐s❡r♠♦s s❛❜❡r ❛ q✉❛♥t✐❞❛❞❡ ❞❡ ♣❛♣❡❧ ❛♣r♦①✐♠❛❞♦ ♣♦❞❡♠♦s s✉❜st✐t✉✐r √3 ≈ 1,73✳
❆ss✐♠✱ t❡♠♦s✿
A = 400√3 = 400·1,73 = 692cm2.
▲♦❣♦ ♣❛r❛ ❢❛③❡r ❛s q✉❛tr♦ ❜❛♥❞❡✐r✐♥❤❛s ✉♠❛ ♣❡ss♦❛ ♣r❡❝✐s❛ ❞❡ ❛♣r♦①✐♠❛❞❛♠❡♥t❡ 692cm2
❞❡ ♣❛♣❡❧✳
✷✳✶✳ ➪❘❊❆ ❉❊ ❋■●❯❘❆❙ P▲❆◆❆❙
✷✳✶✳✻ ➪r❡❛ ❞♦ tr❛♣é③✐♦
❱❛♠♦s ❞❡❝♦♠♣♦r ♦ tr❛♣é③✐♦M N P Q ❡♠ ❞♦✐s tr✐â♥❣✉❧♦s tr❛ç❛♥❞♦ ✉♠❛ ❞❡ s✉❛s ❞✐❛❣♦✲ ♥❛✐s✱ ♣♦✐s ❥á ❝♦♥❤❡❝❡♠♦s ❝♦♠♦ s❡ ❝❛❧❝✉❧❛ ❛ ár❡❛ ❞❡ ✉♠ tr✐â♥❣✉❧♦✳ ❯♠ tr✐â♥❣✉❧♦ t❡♠ ❜❛s❡ B ❡ ❛❧t✉r❛ h ❡ ♦ ♦✉tr♦ t❡♠ ❜❛s❡ b ❡ ❛❧t✉r❛ h✳
❈♦♠♦ ❥á s❛❜❡♠♦s ❝❛❧❝✉❧❛r ❛ ár❡❛ ❞♦ tr✐â♥❣✉❧♦✱ ❜❛st❛ ❢❛③❡r♠♦s✿
➪r❡❛ ❞♦ tr❛♣é③✐♦= B·h 2 +
b·h
2 =
B ·h+b·h
2 =
(B+b)·h
2 .
➪r❡❛ ❞♦ tr❛♣é③✐♦ ❂ (B+b)·h
2 ,
♦♥❞❡ B❂ ❇❛s❡ ♠❛✐♦r✱ b ❂ ❜❛s❡ ♠❡♥♦r ❡ h ❂ ❛❧t✉r❛✳
❆ ár❡❛ ❞❡ ✉♠ tr❛♣é③✐♦ é ✐❣✉❛❧ ❛♦ ♣r♦❞✉t♦ ❞❛ ♠❡t❛❞❡ ❞❛ s♦♠❛ ❞❛s ❜❛s❡s ♣❡❧❛ ❛❧t✉r❛✳
❊①❡♠♣❧♦✿ ▼❛♥♦❡❧ q✉❡r ♣❧❛♥t❛r ❣r❛♠❛ ❡♠ ✉♠❛ ♣❛rt❡ ❞❡ s✉❛ ❝❤á❝❛r❛ q✉❡ t❡♠ ♦ ❢♦r♠❛t♦ ❡ ❞✐♠❡♥sõ❡s ♠♦str❛❞❛s ♥❛ ✜❣✉r❛ ❛❜❛✐①♦✳ ◗✉❛♥t♦s ♠❡tr♦s q✉❛❞r❛❞♦s ❞❡ ❣r❛♠❛ ❡❧❡ ♣r❡❝✐s❛ ❝♦♠♣r❛r❄
❘❡s♦❧✉çã♦✿ ❖❧❤❛♥❞♦ ❛ ✜❣✉r❛ ❛❝✐♠❛ ❞❡ ✐♠❡❞✐❛t♦ ♣❛r❡❝❡ ♥ã♦ t❡r ❝♦♠♦ ❝❛❧❝✉❧❛r s✉❛ ár❡❛✱ ♠❛s ❞❡♣♦✐s ♣❡r❝❡❜❡♠♦s q✉❡ ♣♦❞❡♠♦s ❞✐✈✐❞✐✲❧❛ ❡♠ ❢♦r♠❛s ❝♦♥❤❡❝✐❞❛s ❝♦♠♦ ♠♦str❛ ❛ ✐♠❛❣❡♠ ❛ s❡❣✉✐r✳
