OBMEP 2022 - QUESTÃO 19 O retângulo ABCD é formado pelos triângulos ABP, CQP, AQD e APQ. As áreas dos triângulos ABP, CQP e AQD são, respect...
OBMEP 2022 - QUESTÃO 19
O retângulo ABCD é formado pelos triângulos ABP, CQP, AQD e APQ. As áreas dos triângulos ABP, CQP e AQD são, respectivamente, 6, 5 e 2 cm². Qual é, em em, a área do triângulo APQ?
(A) 7
(B) 8
(C) 9
(D) 10
(E) 11
QUESTÃO ANTERIOR:
GABARITO:
(E) 11
RESOLUÇÃO:
Se AD = BC = x e AB = CD = y, temos que a área do triângulo
, logo,
Assim, ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQskHWg4wqw01zlQIeXXxqIX1KgZqz6IoEAl5gqxrWUK-G0aODHmugxANmzp7z0Fn07eJRgybq5DGRVfs14Btbt0bBYmbZFWndM76yv-UhJFHURv41BMTXlxXZj9NKOaSow3IQtzCKwFZ0y3btJ4-tfP66nemR3sv9JF4KrY7i8TSXGh9ZdonrcL1NYg/s1600/obmep-2022-inagacao-75.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEiOR9wrCXL5wJO5PYxrkfL5gFuiJzDBbJLoHdwW35_F_9wVs0MU0-BrJiA1CgoP1q0CyWSEnxSzYMJg2oIIOMVIghUT-Hb3FMCMk1IGpUPnLbmsl1JRmOrp5xddeFclDzeIlRVrMr-uMpRoLJBuXdEL_nyT0pbIYl0Y2Pqm_5fx4VYdxO93UGHEY52sQg/s1600/obmep-2022-inagacao-73.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhJy7LKj5iSC-uIBw8utXGSBPAyKQp4sWkvqjVQ0bkOXzg7wEA1_zO36fNiDtdI1m99K-v32rxMkxUx1nTr5YdidwXamGz_QvjWs7B1696S93ez8W9OIy9OrjswIhZmSOFOwJDZqvxfn69fh-DPv_KTIM3KHQqoU2Xfo2JfkVaBKf2XnIMDz1J-pje-Kg/s1600/obmep-2022-inagacao-74.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQskHWg4wqw01zlQIeXXxqIX1KgZqz6IoEAl5gqxrWUK-G0aODHmugxANmzp7z0Fn07eJRgybq5DGRVfs14Btbt0bBYmbZFWndM76yv-UhJFHURv41BMTXlxXZj9NKOaSow3IQtzCKwFZ0y3btJ4-tfP66nemR3sv9JF4KrY7i8TSXGh9ZdonrcL1NYg/s1600/obmep-2022-inagacao-75.jpg)
Temos também que a área do triângulo AQD é 2:
, logo,
e![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQLFNq9R491whnNoTcBNHfsJfnfuxKW_NIoIx54AL1FGaj3olIkIxw6Qh8RNlQI8Qr4NGbXaceYqPEwAPCELb1Wq48Zm3SPtkskIkarpYlyTBZUFQq7mJCvkw_j-Ps_e5XipkJWAIXQ0JEcafnkNZeFvT_Fw4KXrVhOmJuZtAD12jaUqxV8de2-pfl1A/s1600/obmep-2022-inagacao-78.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgDNWcVvR8ET2zJrjaW9AY6pC55WM54PuB--tAyzFtOhORfHmBm8LRyT16u1uTjqdDlmjfZvFRzosABXKGojvjWdFWW0eL4l1GO7nqzmVeUCyueaxG_HxVU9JkyjhZP3tQn73ah--BiRFZcxpUvEK6WnfIGtzdy7GRN3pafrHyzoD5URUrN8ny_nptgsg/s1600/obmep-2022-inagacao-76.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEj14BBmhu6-pkOgIcEpn2UK_Bbcf7aymdbrweZQTQmo2mwGEmjHLAOOIKjzG9q2xMFi280qi81JD00nv5fiRNdvbqPjkxWcrQ22fzL8xTiEhMPbVEbs0RVqMQlzgfWvaeXSVd9nyU8GYRFhAkMP1NumwERFPd1iSfz9hquNSKOnzTDz3fznyDAyZ-RdAw/s1600/obmep-2022-inagacao-77.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhQLFNq9R491whnNoTcBNHfsJfnfuxKW_NIoIx54AL1FGaj3olIkIxw6Qh8RNlQI8Qr4NGbXaceYqPEwAPCELb1Wq48Zm3SPtkskIkarpYlyTBZUFQq7mJCvkw_j-Ps_e5XipkJWAIXQ0JEcafnkNZeFvT_Fw4KXrVhOmJuZtAD12jaUqxV8de2-pfl1A/s1600/obmep-2022-inagacao-78.jpg)
Assim, a área do triângulo PCQ é ![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrviAnhc9HgxNcOXtRDSyna5OP7hlvAqoAvdCBzAuVyP6cAwg1kLS7Y9cqKe79wCX3S7j52LY6urKp0s4HInavWTFMJz0OP9lYK1_u5sC5ZK_eD7SZyBROpjlluaqi5JSfN5m11n28ALu2bqBALXZpf1laxCu1BYnhAJTMDKeFA4jBZeckvpLQKGA6Zg/s1600/obmep-2022-inagacao-79.jpg)
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjrviAnhc9HgxNcOXtRDSyna5OP7hlvAqoAvdCBzAuVyP6cAwg1kLS7Y9cqKe79wCX3S7j52LY6urKp0s4HInavWTFMJz0OP9lYK1_u5sC5ZK_eD7SZyBROpjlluaqi5JSfN5m11n28ALu2bqBALXZpf1laxCu1BYnhAJTMDKeFA4jBZeckvpLQKGA6Zg/s1600/obmep-2022-inagacao-79.jpg)
Usando-se as igualdades já encontradas,
Desenvolvendo-se, (xy)² - 26xy + 48 = O. Essa última equação tem como raízes
.
![](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgfY1nVy7lK_QnKSQMZElPCOijovKl9clwIS3UxxwBzSG5s2PsIZ8jw0YpUucr-vYNCPMxdTs76LNRIBb-UUBom1fpn43Rnh1TQaEmlYEt2iNUWdfNcgLGNVSpiu85koFt5yF4Xl6dyA_8rgFkQMfeAXvvcfSnV-nuhcR5-mghXtWZ37lvNjwyjGrwNEg/s1600/obmep-2022-inagacao-81.jpg)
Como xy é a área do quadrilátero ABCD, a única raiz que interessa é xy = 24; logo, a área do triângulo APQ é igual a 24 - (6 + 5 + 2) = 11.
PRÓXIMA QUESTÃO:
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