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Giải bài 4.32 trang 71 SGK Toán 10 – Kết nối tri thức

Cho hình vuông ABCD có cạnh a. Khẳng định nào sau đây là đúng? A. AB.BD=45 B. AC.BC=45 và AC.BC=a^2 C. AC.BD= D. BA.BD=- a^2

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Đề bài

Cho hình vuông ABCD có cạnh a. Khẳng định nào sau đây là đúng? A. \(\left( {\overrightarrow {AB} ,\overrightarrow {BD} } \right) = {45^o}\) B. \(\left( {\overrightarrow {AC} ,\overrightarrow {BC} } \right) = {45^o}\) và \(\overrightarrow {AC} .\overrightarrow {BC}  = {a^2}\) C. \(\overrightarrow {AC} .\overrightarrow {BD}  = {a^2}\sqrt 2 \) D. \(\overrightarrow {BA} .\overrightarrow {BD}  =  - {a^2}\)

Phương pháp giải - Xem chi tiết

Tính tích vô hướng bằng công thức: \(\overrightarrow a .\overrightarrow b  = \left| {\overrightarrow a } \right|.\left| {\overrightarrow b } \right|\,\cos \left( {\overrightarrow a ,\overrightarrow b } \right)\).

Lời giải chi tiết

A. Ta có: \(\left( {\overrightarrow {AB} ,\overrightarrow {BD} } \right) = \left( {\overrightarrow {BE} ,\overrightarrow {BD} } \right) = {135^o} \ne {45^o}.\) Vậy A sai.

 

B. Ta có: \(\left( {\overrightarrow {AC} ,\overrightarrow {BC} } \right) = \left( {\overrightarrow {CF} ,\overrightarrow {CG} } \right) = {45^o}\) và  \(\overrightarrow {AC} .\overrightarrow {BC}  = AC.BC.\cos {45^o} = a\sqrt 2 .a.\frac{{\sqrt 2 }}{2} = {a^2}.\)Vậy B đúng.

 

Chọn B

C. Dễ thấy \(AC \bot BD\) nên \(\overrightarrow {AC} .\overrightarrow {BD}  = 0 \ne {a^2}\sqrt 2.\) Vậy C sai.

 

D. Ta có: \(\left( {\overrightarrow {BA} .\overrightarrow {BD} } \right) = {45^o}\) \( \Rightarrow \overrightarrow {BA} .\overrightarrow {BD}  = BA.BD.\cos {45^o} = a.a\sqrt 2 .\frac{{\sqrt 2 }}{2} = {a^2} \ne  - {a^2}.\) Vậy D sai.

 

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