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Giải bài 4.28 trang 51 sách bài tập toán 9 - Kết nối tri thức tập 1

Chứng minh nếu một góc nhọn của một tam giác vuông có số đo gấp đôi số đo góc nhọn kia thì tam giác đó có một cạnh dài gấp đôi một trong hai cạnh còn lại.

Tổng hợp Đề thi vào 10 có đáp án và lời giải

Toán - Văn - Anh
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Đề bài

Chứng minh nếu một góc nhọn của một tam giác vuông có số đo gấp đôi số đo góc nhọn kia thì tam giác đó có một cạnh dài gấp đôi một trong hai cạnh còn lại.

Phương pháp giải - Xem chi tiết

+ Giả sử tam giác ABC vuông tại A có \(\widehat B = 2\widehat C\). + Tính được góc C của tam giác ABC. + Tam giác ABC vuông tại A nên \(\sin C = \frac{{AB}}{{BC}}\) nên \(BC = 2AB\).

Lời giải chi tiết

Giả sử tam giác ABC vuông tại A có \(\widehat B = 2\widehat C\). Khi đó, \(\widehat B + \widehat C = {90^o}\), suy ra \(2\widehat C + \widehat C = {90^o}\), suy ra \(\widehat C = {30^o}\). Tam giác ABC vuông tại A nên \(\sin C = \frac{{AB}}{{BC}}\), suy ra \(\frac{{AB}}{{BC}} = \sin {30^o} = \frac{1}{2}\). Do đó, \(BC = 2AB\). Vậy nếu một góc nhọn của một tam giác vuông có số đo gấp đôi số đo góc nhọn kia thì tam giác đó có một cạnh dài gấp đôi một trong hai cạnh còn lại.

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