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Giải bài 13 trang 89 sách bài tập toán 9 - Chân trời sáng tạo tập 2

Cho tam giác ABC có (O) là đường tròn ngoại tiếp. Vẽ đường cao AH của tam giác ABC và đường kính AD của đường tròn (O). Biết AB = 8 cm; AC = 15 cm và AH = 5 cm. a) Chứng minh (Delta AHBbacksim Delta ACD). b) Tính độ dài bán kính của đường tròn

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

Toán - Văn - Anh
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Đề bài

Cho tam giác ABC có (O) là đường tròn ngoại tiếp. Vẽ đường cao AH của tam giác ABC và đường kính AD của đường tròn (O). Biết AB = 8 cm; AC = 15 cm và AH = 5 cm. a) Chứng minh \(\Delta AHB\backsim \Delta ACD\). b) Tính độ dài bán kính của đường tròn

Phương pháp giải - Xem chi tiết

Chứng minh theo trường hợp góc – góc. Từ chứng minh phần a suy ra tỉ số đồng dạng và tìm bán kính của đường tròn.

Lời giải chi tiết

a) Ta có \(\widehat {ACD}\) là góc nội tiếp chắn nửa đường tròn 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). b) Vì \(\Delta AHB\backsim \Delta ACD\) nên \(\frac{{AH}}{{AC}} = \frac{{AB}}{{AD}}\) hay AH.AD = AB.AC, suy ra AD = \(\frac{{8.15}}{5} = 24\) (cm). Do đó độ dài bán kính của đường tròn (O) là \(\frac{{24}}{2} = 12\) (cm).

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