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Câu 37 trang 68 SGK Hình học 11 Nâng cao

Cho hình hộp ABCD.A’B’C’D’. Chứng minh rẳng a. mp(BDA’) // mp(B’D’C) b.Đường chéo AC’ đi qua các trọng tâm G1, G2 của hai tam giác BDA’ và B’D’C
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Cho hình hộp ABCD.A’B’C’D’. Chứng minh rẳng

LG a

mp(BDA’) // mp(B’D’C)

Phương pháp giải:

Mặt phẳng (P) chứa hai đường thẳng cắt nhau cùng song song với (Q) thì (P)//(Q).

Lời giải chi tiết:

Chứng minh ( BDA’) // (B’D’C)

Tứ giác BB’D’D và A’B’CD là các hình bình hành nên : BD // B’D’ và DA’ // B’C \(BD//B'D' \subset \left( {B'D'C} \right)\)\( \Rightarrow BD//\left( {B'D'C} \right)\) \(DA'//CB' \subset \left( {B'D'C} \right)\)\( \Rightarrow DA'//\left( {B'D'C} \right)\) Mà \(BD,DA' \subset \left( {A'BD} \right) \)\(\Rightarrow \left( {A'BD} \right)//\left( {B'D'C} \right)\) Vậy (BDA’) // (B’D’C).

LG b

Đường chéo AC’ đi qua các trọng tâm G1, G2 của hai tam giác BDA’ và B’D’C

Lời giải chi tiết:

Chứng minh G1 , G2 ∈ AC’

Gọi O, O’ lần lượt là tâm của hình bình hành ABCD và A’B’C’D’.

Trong mặt phẳng (AA’C’C) gọi G1 , G2 lần lượt là giao điểm của AC’ với A’O và O’C.

Ta chứng minh G1, G2 lần lượt là trọng tâm của ∆A’BD và ∆CB’D’.

Thật vậy, ta có ∆G1OA đồng dạng ∆G1A’C’ ( vì AC // A’C’)

\( \Rightarrow {{{G_1}O} \over {{G_1}A'}} = {{OA} \over {A'C'}} = {1 \over 2} \Rightarrow {{A'{G_1}} \over {A'O}} = {2 \over 3}\)

⇒ G1 là trọng tâm ∆A’BD.

Tương tự, G2 là trọng tâm ∆CB’D’. Vậy AC’ đi qua G1, G2 .

LG c

G1 và G2 chia đoạn AC’ thành ba phần bằng nhau

Lời giải chi tiết:

Chứng minh AG1 = G1G2 = G2C’

Theo câu trên , ta có:

\({{A{G_1}} \over {{G_1}C'}} = {{AO} \over {A'C'}} = {1 \over 2}\) ( vì ∆G1OA đồng dạng ∆G1♔A’C’) \( \Rightarrow A{G_1} = {1 \over 3}AC'\)  (1)

Tương tự: \({{C'{G_2}} \over {{G_2}A}} = {{C'O'} \over {CA}} = {1 \over 2}\) ( vì ∆G2C’O' đồng dạng ∆G2ﷺAC) \( \Rightarrow C'{G_2} = {1 \over 3}AC'\)   (2)

Từ (1) và (2) suy ra: AG1 = G1G2 = G2C’.

LG d

Các trung điểm của sáu cạnh BC, CD, DD’, D’A’, A’B’,B’B cùng nằm trên một mặt phẳng

Lời giải chi tiết:

Gọi M, N, P, Q, S, R lần lượt là trung điểm của các cạnh AB, AD, DD’, C’D’, C’B’, B’B. Ta có: \(\left\{ {\matrix{   {MN//BD}  \cr   {SP//BD}  \cr } } \right. \Rightarrow MN//SP\) Gọi (α) = (MN, SP) Ta có : \(\left\{ {\matrix{   {PQ//DC'}  \cr   {MS//AB'}  \cr } } \right. \Rightarrow PQ//MS\) ( vì DC’ // AB’) ⇒ PQ ⊂ (α) do đó Q ∈ (α). Tương tự: QR // MN ⇒ QR ⊂ (α) do đó R ∈ (α). Vậy M, N, P, Q, R, S ∈ (α). Mặt khác vì \(\left\{ {\matrix{   {MS//AB'}  \cr   {NP//AD'}  \cr } } \right.\)  nên (MNPQRS) // (AB’D').

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