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Bài 2 trang 81 SGK Toán 11 tập 2 – Chân trời sáng tạo

Cho hai tam giác cân (ABC) và (ABD) có đáy chung (AB) và không cùng nằm trong một mặt phẳng.

Tổng👍 hợp đề thi học kì 2 lꦕớp 11 tất cả các môn - Chân trời sáng tạo

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

Cho hai tam giác cân \(ABC\) và \(ABD\) có đáy chung \(AB\) và không cùng nằm trong một mặt phẳng. a) Chứng minh rằng \(AB \bot CD\). b) Xác định đoạn vuông góc chung của \(AB\) và \(C{\rm{D}}\).

Phương pháp giải - Xem chi tiết

‒ Cách chứng minh hai đường thẳng vuông góc:

Cách 1: Chứng minh góc giữa chúng bằng \({90^ \circ }\).

Cách 2: Chứng m🔯inh ꦫđường thẳng này vuông góc với một mặt phẳng chứa đường thẳng kia.

‒ Cách dựng đoạn vuông góc chung của hai đường thẳng \(a\) và \(b\) vuông góc với nhau:

Bước 1: Xác địꦬnh mặt phẳng \(\left( P \right)\) chứa \(b\) mà \(\left( P \right)\) vuông góc với \(a\).

Bước 2: Tìm giao điểm \(I = \left( P \right) \cap a\).

Bước 3: Kẻ \(IA \b🌸ot b\left( {A \in b} \right)\), chứng minh \(IA \bot a\). Khi đó \(d\left( {a,b} \right) = IA\).

Lời giải chi tiết

 

a) Gọi \(I\) là trung điểm của \(AB\)\(\Delta ABC\) cân tại \(C\)\( \Rightarrow CI \bot AB\)\(\Delta ABD\) cân tại \(D\)\( \Rightarrow DI \bot AB\)\( \Rightarrow AB \bot \left( {C{\rm{D}}I} \right) \Rightarrow AB \bot C{\rm{D}}\)b) Kẻ \(IH \bot C{\rm{D}}\left( {H \in C{\rm{D}}} \right)\)\(AB \bot \left( {C{\rm{D}}I} \right) \Rightarrow AB \bot IH\)Vậy \(IH\) là đoạn vuông góc chung của \(AB\) và \(C{\rm{D}}\).

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