OBMEP 2022 - QUESTÃO 12 Na figura, ABCD é um quadrado e AGD, BEC e CDF são triângulos equiláteros. Quanto mede o ângulo GFE? (A) 15° (B) 22,...
OBMEP 2022 - QUESTÃO 12
Na figura, ABCD é um quadrado e AGD, BEC e CDF são triângulos equiláteros. Quanto mede o ângulo GFE?
(A) 15°
(B) 22,5°
(C) 30°
(D) 36°
(E) 45°
QUESTÃO ANTERIOR:
GABARITO:
(C) 30°
RESOLUÇÃO:
Observemos primeiro os ângulos com vértice D. Como o triângulo DFC é equilátero, temos ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjguzjFRPJzRhOR_vgbvCEfRZbbI8YCLYHErcoZ-GTc3YJvMldbU-FNpX_ynDMY2xzjJhLXKZOMJR84o7hcDkrXFBu5rBRGvvR62thmB9L-WxV7iuscUyuQCD2uA0b0cohK79hXdTlLj2g4ObZlJt0SvFVTeBlkTAgPqjmp_7UcX0YYgBfhj1SwbSjR9Q/s1600/obmep-2022-inagacao-42.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjguzjFRPJzRhOR_vgbvCEfRZbbI8YCLYHErcoZ-GTc3YJvMldbU-FNpX_ynDMY2xzjJhLXKZOMJR84o7hcDkrXFBu5rBRGvvR62thmB9L-WxV7iuscUyuQCD2uA0b0cohK79hXdTlLj2g4ObZlJt0SvFVTeBlkTAgPqjmp_7UcX0YYgBfhj1SwbSjR9Q/s1600/obmep-2022-inagacao-42.jpg)
e, como
segue que ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaI4NklnvuRr5UpujfwDiucaKz46v2fuhu63xQFCF4V5WLVNBBmA2XknViQhggWleSKReM50DmKbnp79wUVDg6HVk_0QMuD4D9N78Pa9QTEG1xAUjfAIaW1hsxMjZRWQcldYBWjEH9QuhJH50bXgw6F6vAfhqbsgwgGocgyGdjgotdhn1PpirJ6bNDVg/s1600/obmep-2022-inagacao-44.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEju-pWuwgn33P8xjKFR7kemMUQmS531TFDJy4CH0sdjDKDq76d-gZxgYZZCbSbjZPE8mxKB6CVAu48XvavSXVT1kpZe133ZShOvqr-W9mpy4gHD-SwViHR_R-Y6eXlQcJx4rrIN7B4FNPPy0sHlG2xWGb3krjzAF_NFpyeRdUFXR59gIBLDBwic2ITjWw/s1600/obmep-2022-inagacao-43.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiaI4NklnvuRr5UpujfwDiucaKz46v2fuhu63xQFCF4V5WLVNBBmA2XknViQhggWleSKReM50DmKbnp79wUVDg6HVk_0QMuD4D9N78Pa9QTEG1xAUjfAIaW1hsxMjZRWQcldYBWjEH9QuhJH50bXgw6F6vAfhqbsgwgGocgyGdjgotdhn1PpirJ6bNDVg/s1600/obmep-2022-inagacao-44.jpg)
Analogamente temos
e então ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgW470uB-SyKKXJGsr7FhRXCVhERaZ8V8mW-zIOoIT94TjVA8SbeLmWbZX9xX2BE04hQ5Rc8AYzQ1p_bDw5FFnFKmExlS86NTy05IyTzFpdevseUrKSx2l5Nm7EFQXDZpwCck8tGb0ItOcKhHc9oE7oH722J17LqFKNDBAMTFsT5MF1KXmXq4RUPH_oQ/s1600/obmep-2022-inagacao-46.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjPEmQKQA_DJ8CmMsCV55f78vZhgZzuRFSn3EGmgszluCsypfoc03tOa0g3dERFANR7BzDlVNbp894EGAO9ozdoPkYhZU407oTUEq0lI-bBgQmdUi9_PucYYpzhAkNJxs6JRDg_lLyDIfgGbOr2XMSr9oaQmeeJTQFw9-XjScVyYZg6IQrUOKffwrcTOQ/s1600/obmep-2022-inagacao-45.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjgW470uB-SyKKXJGsr7FhRXCVhERaZ8V8mW-zIOoIT94TjVA8SbeLmWbZX9xX2BE04hQ5Rc8AYzQ1p_bDw5FFnFKmExlS86NTy05IyTzFpdevseUrKSx2l5Nm7EFQXDZpwCck8tGb0ItOcKhHc9oE7oH722J17LqFKNDBAMTFsT5MF1KXmXq4RUPH_oQ/s1600/obmep-2022-inagacao-46.jpg)
Passemos ao triângulo GDF. Ele é isósceles, pois DG = DF, e segue que ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxrRjVneq_khj6nrDXbReRcqL076T1VPff64zftl-3qbeR7NjhIl6I_qJ9xwXA1uc8ovsv94ZpEw2EFBaFf3ROhewfw7yZjCka3bAjEtIZkPquDae-On6ubKmIorDeNgYVIBxsyzOUaFBpa26YA6mBipxYJRhRXt7UVdAvI9EHw-e7J99k35-AEMHsew/s1600/obmep-2022-inagacao-47.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgxrRjVneq_khj6nrDXbReRcqL076T1VPff64zftl-3qbeR7NjhIl6I_qJ9xwXA1uc8ovsv94ZpEw2EFBaFf3ROhewfw7yZjCka3bAjEtIZkPquDae-On6ubKmIorDeNgYVIBxsyzOUaFBpa26YA6mBipxYJRhRXt7UVdAvI9EHw-e7J99k35-AEMHsew/s1600/obmep-2022-inagacao-47.jpg)
Logo ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSOQK2HuV1NEhWJu3iUdsAv897FW9RrJuEOozVSxtZvQTeaL_JZhwm0YC6_oRPTS27NoVkOlaDLPZdyOwWCGZSsQMc_C57jc2VJ_DOsQRA_0P2D31rGADcaVFUfF1FaJfpgqqUghJnymr4M_P7PYJvU2cAlw302SBQXitfwl5QmDb8vmwEj28u-HweuQ/s1600/obmep-2022-inagacao-48.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLS48PEIghUrcdmELbQav4eM4j-IfwlT2HnFsXi611F6TBRF8liSLGm8buywo1C_TZYci31pUp1dqOy4fVV83uYMWvT-fhkvtlxAKGwS5aYme3bcBn4P3gUzMgQH9bI9DZFN7lrdsq17Pe34foO7XUJMcOqORTZNwO8V98Z6lH8YA1s82CE4RIJHxnFA/s1600/obmep-2022-inagacao-49.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhSOQK2HuV1NEhWJu3iUdsAv897FW9RrJuEOozVSxtZvQTeaL_JZhwm0YC6_oRPTS27NoVkOlaDLPZdyOwWCGZSsQMc_C57jc2VJ_DOsQRA_0P2D31rGADcaVFUfF1FaJfpgqqUghJnymr4M_P7PYJvU2cAlw302SBQXitfwl5QmDb8vmwEj28u-HweuQ/s1600/obmep-2022-inagacao-48.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjLS48PEIghUrcdmELbQav4eM4j-IfwlT2HnFsXi611F6TBRF8liSLGm8buywo1C_TZYci31pUp1dqOy4fVV83uYMWvT-fhkvtlxAKGwS5aYme3bcBn4P3gUzMgQH9bI9DZFN7lrdsq17Pe34foO7XUJMcOqORTZNwO8V98Z6lH8YA1s82CE4RIJHxnFA/s1600/obmep-2022-inagacao-49.jpg)
Agora consideremos o triângulo CFE. Como acima, temos
e segue que ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikhS-MQchX7mwbCPVNkay-fNYG6U2ODJy6noHbjR8wUyMpHqTlVCNAA2uZwpg6TLkixQ80tp6WKxYjdwGjWvtPbH11T-Hv8_VgcoZqf13Gsp4oFYs24EZMddXA8jBOC1hXUKy1Zb5X2XJ3Z9iOJsEHkQP009F5xCJqlthiBE9DRuST8ExhCBuqVfsS4Q/s1600/obmep-2022-inagacao-51.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiBFDDTpMvTEe1k7U8Mparp4PnxBH57oORzQmqoT45o3VUZpwxi7rNJgfWw13zY0AfYZvOARRu19lYmdVXz5JSHwtS1DBmoJPUV-DasApdCyzgOgh6IsRLuYH3APmPsX390QIXNwfEdPn1qzYvnaZMPqjtZ71ltN5hOg36Ree_KwY2bkIxf-aGIEsPpww/s1600/obmep-2022-inagacao-50.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEikhS-MQchX7mwbCPVNkay-fNYG6U2ODJy6noHbjR8wUyMpHqTlVCNAA2uZwpg6TLkixQ80tp6WKxYjdwGjWvtPbH11T-Hv8_VgcoZqf13Gsp4oFYs24EZMddXA8jBOC1hXUKy1Zb5X2XJ3Z9iOJsEHkQP009F5xCJqlthiBE9DRuST8ExhCBuqVfsS4Q/s1600/obmep-2022-inagacao-51.jpg)
Por outro lado, como CF = CE, esse triângulo também é isósceles, e temos, então, ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh02vmkCwcQklzmzBACegKZ3Fbd6A4DMUYv_Y2UY7diGO8GR24e-v3UYuAVtuMTmBUK9zsw2vyzP47sKT4bAPCa-d5q1mm0GP5OMXt61KMRQo4ka2hos1RFLf5UeQDkEg2E1faydWuLKWMzstYWrEQVSbdJPIFHVhMW3WNlhkneNSEeOl6Pwak7rAw9ag/s1600/obmep-2022-inagacao-52.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEh02vmkCwcQklzmzBACegKZ3Fbd6A4DMUYv_Y2UY7diGO8GR24e-v3UYuAVtuMTmBUK9zsw2vyzP47sKT4bAPCa-d5q1mm0GP5OMXt61KMRQo4ka2hos1RFLf5UeQDkEg2E1faydWuLKWMzstYWrEQVSbdJPIFHVhMW3WNlhkneNSEeOl6Pwak7rAw9ag/s1600/obmep-2022-inagacao-52.jpg)
Finalmente, temos ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuGiWzDsDbNsrw-eHzm9V4UsOUz5lnB7VhQch87XVIwZuxCFzLKA6ff3wYDQ2eXJ-I43KDUGOWHQN8mJOlmJzo9_dRMnb6k9-_KxoFXJp_4y6NkM_LvHmE5GYod9-5pdI5jrV52ndtB9ttcNCgWwmvqjOxXnfRNtFv7K7SIOw6WjCJNbFFh0q6cl_SMA/s1600/obmep-2022-inagacao-53.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiuGiWzDsDbNsrw-eHzm9V4UsOUz5lnB7VhQch87XVIwZuxCFzLKA6ff3wYDQ2eXJ-I43KDUGOWHQN8mJOlmJzo9_dRMnb6k9-_KxoFXJp_4y6NkM_LvHmE5GYod9-5pdI5jrV52ndtB9ttcNCgWwmvqjOxXnfRNtFv7K7SIOw6WjCJNbFFh0q6cl_SMA/s1600/obmep-2022-inagacao-53.jpg)
OBMEP 2022 - QUESTÃO 13
DISPONÍVEL NA PROVA NÍVEL 1.
OBMEP 2022 - QUESTÃO 14
DISPONÍVEL NA PROVA NÍVEL 1.
PRÓXIMA QUESTÃO:
QUESTÃO DISPONÍVEL EM: