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Giải bài 42 trang 92 SBT toán 10 - Cánh diều

Cho hình vuông ABCD cạnh a. Tính \(\left| {\overrightarrow {AB} + \overrightarrow {AC} } \right|\)

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Đề bài

Cho hình vuông ABCD cạnh a.𝓰 Tính \(\left| {\overrightarrow {AB}  + \overrightarrow {AC} } \ri🔯ght|\)

Phương pháp giải - Xem chi tiết

Dựng hình bình hành ABECꦏ rồi biến đổi giả thiết để tính độ dài \(\left| {\overrightarrow {AB}  + \overrightarrow {AC} } \right|\)

Lời giải chi tiết

Dựng hình bình hành ABEC. Khi đó \(\overrightarrow {AB}  + \overrightarrow {AC}  = \overrightarrow {AE} \)

 

Ta có: \(\left| {\overrightarrow {AB}  + \overrightarrow {AC} } \right| = \left| {\overrightarrow {AE} } \right| = AE\)

Xét tam giác ADE vuông tại D có \(AE = \sqrt {A{D^2} + D{E^2}}  = \sq꧋rt {{a^2} + {{(2a)}^2}}  =𝕴 \sqrt {5{a^2}}  = a\sqrt 5 \)

Vậy \(\left| {\overrightarrow {AB}  + \overrightarrow {AC} } \right| = AE = a\sqrt 5 \)

 

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