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Giải bài 4 trang 65 SGK Toán 8 – Cánh diều

Cho tam giác ABC nhọn có H là trực tâm. Gọi M, N, P, Q lần lượt là trung điểm của các đoạn thẳng

꧅Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên
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Đề bài

Cho tam giác ABC nhọn có H là trực tâm. Gọi M, N, P, Q lần lượt là trung điểm của các đoạn thẳng AB, BH, HC, CA. Chứng minh tứ giác MNPQ là hình chữ nhật.

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Sử dụng định lý đường trung bình và dấu hiệu nhận biết của hình chữ nhật để chứng minh bài toán

Lời giải chi tiết

Vì M, N lần lượt là trung điểm của các đoạn thẳng AB, BH nên ta có: MN là đường trung bình tam giác ABH \( \Rightarrow MN//AH\) mà \(AH \bot BC\) nên \(MN \bot BC\) (1) Vì P, Q lần lượt là trung điểm của các đoạn thẳng CH, AC nên ta có: PQ là đường trung bình tam giác AHC \( \Rightarrow PQ//AH\) mà \(AH \bot BC\) nên \(QP \bot BC\) (2) Vì P, N lần lượt là trung điểm của các đoạn thẳng CH, BH nên ta có: PN là đường trung bình tam giác BHC \( \Rightarrow PN//BC\) mà \(AH \bot BC\) nên \(PN \bot AH\)(3) Vì M, Q lần lượt là trung điểm của các đoạn thẳng AB, AC nên ta có: MQ là đường trung bình tam giác ABC \( \Rightarrow MQ//BC\) mà \(AH \bot BC\) nên \(MQ \bot AH\)(4) Từ (1), (2), (3), (4) ta có \(\widehat {MNP} = \widehat {NPQ} = \widehat {PQM} = \widehat {QMN} = 90^\circ \) Vậy tứ giác MNPQ là hình chữ nhật (dhnb).

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