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Giải bài 8 trang 95 SGK Toán 8 – Cánh diều

Cho Hình 105. Chứng minh:

𓆏Tổng 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 Hình 105. Chứng minh:

a) \(\Delta HAB \backsim \Delta HBC\) b) \(HB = HD = 6cm\)

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

a) Chứng minh \(\widehat {HBC} = \widehat {BAH}\) rồi chứng minh hai tam giác đồng dạng. b) Tính độ dài HB và HD rồi so sánh với nhau.

Lời giải chi tiết

a) Xét tam giác ABC vuông tại B có: \(\widehat {BAC} + \widehat {BCA} = 90^\circ \) Xét tam giác BHC vuông tại H có: \(\begin{array}{l}\widehat {HBC} + \widehat {HCB} = 90^\circ \\ \Rightarrow \widehat {HBC} + \widehat {BCA} = 90^\circ \end{array}\) \( \Rightarrow \widehat {HBC} = \widehat {BAC}\) hay \(\widehat {HBC} = \widehat {BAH}\) Xét tam giác HAB và tam giác HBC có: \(\widehat {BAH} = \widehat {CBH}\) và \(\widehat {BHA} = \widehat {CHB} = 90^\circ \) \( \Rightarrow \Delta HAB \backsim \Delta HBC\) b) Vì \(\Delta HAB \backsim \Delta HBC\) nên \(\begin{array}{l}\frac{{HA}}{{HB}} = \frac{{HB}}{{HC}}\\ \Rightarrow H{B^2} = HA.HC\\ \Rightarrow H{B^2} = 4.9 = 36\\ \Rightarrow HB = 6cm\end{array}\) Ta chứng minh được \(\Delta HAD \backsim \Delta HDC\) \(\begin{array}{l} \Rightarrow \frac{{HA}}{{HD}} = \frac{{HD}}{{HC}}\\ \Rightarrow H{D^2} = HA.HC\\ \Rightarrow H{D^2} = 4.9 = 36\\ \Rightarrow HD = 6cm\end{array}\) Vậy \(HB = HD = 6cm\).

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