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Giải bài 19 trang 95 sách bài tập toán 11 - Cánh diều

Cho hình tứ diện ABCD có \(AB \bot \left( {BCD} \right),\)các tam giác BCD và ACD

൩Tổng hợp đề thi học kì 2 lớp 11 tất cả các môn - Cánh diều

Toán - Văn - Anh - Lí - Hóa - Sinh
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Đề bài

Cho hình tứ diện ABCD có \(AB \bot \left( {BCD} \right),\)các tam giác BCD và ACD là những tam giác nhọn. Gọi H, K lần lượt là trực tâm của các tam giác BCD, ACD. Chứng minh rằng: a) \(AD \bot CH;\)                        b*) \(HK \bot \left( {ACD} \right).\)

Phương pháp giải - Xem chi tiết

Nếu một đường thẳng vuông góc với hai đường thẳng cắt nhau cùng thuộc một mặt phẳng thì nó vuông góc với mặt phẳng ấy.

Lời giải chi tiết

a) Vì \(AB \bot \left( {BCD} \right),{\rm{ }}CH \subset \left( {BCD} \right) \Rightarrow AB \bot CH.\) Do H là trực tâm của tam giác (BCD) nên \(CH \bot BD.\) Mà AB, BD cắt nhau trong mặt phẳng (ABD) nên \(CH \bot \left( {ABD} \right).\)Từ \(CH \bot \left( {ABD} \right),{\rm{ }}AD \subset \left( {ABD} \right) \Rightarrow AD \bot CH.\)b*) Vì H là trực tâm của tam giác BCD nên \(BH \bot CD.\)Lại có, \(AB \bot \left( {BCD} \right),{\rm{ }}CD \subset \left( {BCD} \right) \Rightarrow AB \bot CD.\)Mà AB, BD cắt nhau trong mặt phẳng (ABI) nên \(CD \bot \left( {ABI} \right).\)Từ \(CD \bot \left( {ABI} \right),{\rm{ }}HK \subset \left( {ABI} \right) \Rightarrow CD \bot HK.\)Vì K là trực tâm của tam giác ACD nên \(CK \bot AD.\) Mà CK, CH cắt nhau trong mặt phẳng (CHK) nên \(AD \bot \left( {CHK} \right).\)Lại có, \(AD \bot \left( {CHK} \right),{\rm{ }}HK \subset \left( {CHK} \right) \Rightarrow AD \bot HK.\)Bên cạnh đó, AD, CD cắt nhau trong mặt phẳng (ACD) nên \(HK \bot \left( {ACD} \right).\)

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