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Giải bài tập 5.34 trang 127 SGK Toán 9 tập 1 - Cùng khám phá

Trong Hình 5.73, bốn cạnh của tứ giác ABCD tiếp xúc với đường tròn (O). Chứng minh rằng \(AD + BC = AB + CD\).

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Đề bài

Trong Hình 5.73, bốn cạnh của tứ giác ABCD tiếp xúc với 𓆉đường tròn (O). Chứng minh rằng \(AD + BC = AB + CD\).

Phương pháp giải - Xem chi tiết

+ Chứng minh AD, AB, BC, CD là các tiếp tuyến của (O). + Sử dụng tính chất của hai tiếp tuyến cắt nhau để chỉ ra \(AQ = AM\), \(BM = BN\), \(CN = CP\), \(QD = DP\). + Ta có: \(AD + BC\)\( = AQ + DQ + BN + NC\)\( = AM + DP + BM + PC\)\( = AB + DC\).

Lời giải chi tiết

Vì AD tiếp xúc với (O) tại Q nên AD là tiếp tuyến của (O) với Q là tiếp điểm. Tương tự ta có: AB, BC, CD là các tiếp tuyến của (O). Vì AQ và AM là tiếp tuyến của (O) nên \(AQ = AM\). Tương tự ta có: \(BM = BN\), \(CN = CP\), \(QD = DP\). Ta có: \(AD + BC\)\( = AQ + DQ + BN + NC\)\( = AM + DP + BM + PC\)\( = \left( {AM + BM} \right) + \left( {DP + PC} \right)\)\( = AB + DC\).

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