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Giải bài 8 trang 99 vở thực hành Toán 8 tập 2

Cho hình vuông ABCD có cạnh bằng 4cm và M, N lần lượt là trung điểm của AB, BC. Gọi O là giao điểm của CM và DN (H.9.17). Tính độ dài đoạn thẳng OM.

Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Kết nối t🐟ri thức

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

Cho hình vuông ABCD có cạnh bằng 4cm và M, N lần lượt là trung điểm của AB, BC. Gọi O là giao điểm của CM và DN (H.9.17). Tính độ dài đoạn thẳng OM.

Phương pháp giải - Xem chi tiết

Chứng minh CM $\bot $ DN. Tính diện tích tam giác DMN => OM.

Lời giải chi tiết

Ta có $\Delta CBM=\Delta DCN$(hai tam giác vuông có hai cặp cạnh tương ứng bằng nhau).Do đó $\widehat{CNO}+\widehat{NCO}=\widehat{CND}+\widehat{BCM}=\widehat{BMC}+\widehat{BCM}={{90}^{0}}$.

Vì tổng các góc trong tam giác NOC bằng 1800 nê🦋n: $\widehat{NOC}={{180}^{0}}-\widehat{CNO}-\widehat{N💜CO}={{90}^{0}}$.

Suy ra CM vuông góc với DN.Gọi S là diện tích hình vuông ABCD. Ta có:${{S}_{\Delta DMN}}=S-{{S}_{\Delta NBM}}-{{S}_{\Delta MAC}}=16-4-2-4=6(c{{m}^{2}})$.Do vậy OM. DN = $2{{S}_{\Delta DMN}}$. Suy ra $OM=\frac{2{{S}_{\Delta DMN}}}{\sqrt{C{{D}^{2}}+C{{N}^{2}}}}=\frac{12}{\sqrt{16+4}}=\frac{6\sqrt{5}}{5}(cm)$.

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