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Giải bài 2 trang 79 sách bài tập toán 9 - Chân trời sáng tạo tập 2

Cho tam giác ABC có đường cao AH (H( in )BC) và nội tiếp đường tròn (O). Vẽ đường kính AD của đường tròn (O). Chứng minh AB.AC = AH.AD

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

Toán - Văn - Anh
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Đề bài

ಞCho tam giác ABC có đường cao AH (H\( \in \)BC) và nội tiếp đường tròn (O). Vẽ đường kính AD của đường tròn (O). Chứng minh AB.AC = AH.AD

Phương pháp giải - Xem chi tiết

Chứng minh \(\Delta \)AHB và \(\Delta \)ACD đồng dạng với nhau rồi suy ra tỉ số đồng dạng.

Lời giải chi tiết

Ta có \(\widehat {ACD}\) là góc nội tiếp chắn nửa đường tròn (O) nên \(\widehat {ACD} = {90^o}\). Xét \(\Delta \)AHB và \(\Delta \)ACD có \(\widehat {AHB} = \widehat {ACD} = {90^o}\) \(\widehat {ABH} = \widehat {ADC}\) (hai góc nội tiếp cùng chắn cung AC). Do đó \(\Delta \)AHB \(\backsim \) \(\Delta \)ACD (g.g). Suy ra \(\frac{{AB}}{{AD}} = \frac{{AH}}{{AC}}\) hay AB.AC = AH. AD.

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Tham Gia Group Dành Cho Lớp 9 - Ôn Thi Vào Lớp 10 Miễn Phí

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