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Giải bài tập 4.3 trang 82 SGK Toán 9 tập 1 - Cùng khám phá

Cho hình chữ nhật ABCD có \(\widehat {ABD} = 2\widehat {CBD}\). Hãy tính tỉ số chiều dài và chiều rộng của hình chữ nhật ABCD.

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Đề bài

Cho hình chữ nhật ABCD có \(\widehat {ABD} = 2\widehat {CBD}\). Hãy tính tỉ số chiều dài và chiều rộng của hình chữ nhật ABCD.

Phương pháp giải - Xem chi tiết

+ Vì ABCD là hình chữ nhật nên \(\widehat A = \widehat {ABC} = {90^o}\), kết hợp với \(\widehat {ABD} = 2\widehat {CBD}\) nên tính được góc ABD. + Tam giác ABD vuông tại A nên \(\tan ABD = \frac{{AD}}{{AB}}\), từ đó suy ra tỉ số chiều dài và chiều rộng.

Lời giải chi tiết

Vì ABCD là hình chữ nhật nên \(\widehat A = \widehat {ABC} = {90^o}\). Lại có: \(\widehat {ABD} = 2\widehat {CBD}\) nên \(\widehat {ABD} + \frac{1}{2}\widehat {ABD} = {90^o}\), suy ra \(\widehat {ABD} = {90^o}:\frac{3}{2} = {60^o}\). Tam giác ABD vuông tại A nên \(\frac{{AD}}{{AB}} = \tan ABD = \tan {60^o} = \sqrt 3 \). Do đó, tỉ số chiều dài và chiều rộng của hình chữ nhật ABCD là \(\sqrt 3 \).

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