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Giải bài 8 trang 81 vở thực hành Toán 7

Bài 8. Tam giác ABC vuông tại đỉnh A và có \(\widehat B = {30^o}\). Chứng minh rằng BC = 2AC

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Đề bài

Bàiꦇ 8. Tam giác ABC vuông tại đỉnh A và có \(\widehat B = {30^o}\). Chứng minh rằng BC = 2AC

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Lời giải chi tiết

GT \(\Delta ABC\)vuông tại A, \(\widehat B = {30^o}\)
KL BC = 2AC
Trên tia đối của tia AC lấy điểm D sao cho AD = AC.Tam giác ACB và tam giác ADB vuông tại A và có:AB là cạnh chungAC = AD (theo cách dựng)Vậy \(\Delta ACB = \Delta ADB\)(hai cạnh góc vuông). Do đó BC = BDVậy tam giác BCD cân tại B. Suy ra \(\widehat {ABD} = \widehat {ABC} = {30^o}\)Như vậy:\(\begin{array}{l}\widehat {CBD} = \widehat {ABC} + \widehat {ABD} = {60^o}\\\widehat {CDB} = \widehat {DCB} = \frac{{{{180}^o} - {{60}^o}}}{2} = {60^o}\end{array}\)Do vậy ABD là tam giác đều. Do đó BC = DC = 2AC
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