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Cách chứng minh hai đoạn thẳng bằng nhau, hai đoạn thẳng song song, vuông góc - Toán 9

1. Cách chứng minh hai đoạn thẳng bằng nhau, hai đoạn thẳng song song, vuông góc

Vận dụng kiến thức về: - Tiếp tuyến, tính chất hai tiếp tuyến cắt nhau - Hệ thức lượng trong tam giác vuông - Định lí Pythagore

2. Tính chất của hai tiếp tuyến cắt nhau

Nếu hai tiếp tuyến của một đường tròn cắt nhau tại một điểm thì:
• Điểm đó cách đều hai tiếp điểm.
• Tia kẻ từ điểm đó đi qua tâm là tia phân giác của góc tạo bởi hai tiếp tuyến.
• Tia kẻ từ tâm đi qua điểm đó là tia phân giác của góc tạo bởi hai bán kính đi qua các tiếp điểm.

Ví dụ: Cho đường tròn (O), B, C \( \in \) (O). Tiếp tuyến của (O) tại B và C cắt nhau tại A.

Khi đó: - AB = AC - Tia AO là tia phân giác của \(\widehat {BAC}\). - Tia OA là tia phân giác của \(\widehat {BOC}\).

3. Cách tính cạnh góc vuông theo cạnh huyền và tỉ số lượng giác của góc nhọn

Trong tam giác vuông, mỗi cạnh góc vuông bằng cạnh huyền nhân với sin góc đối hoặc nhân với côsin góc kề.

Cạnh góc vuông = (cạnh huyền ) × (sin góc đối)

= (cạnh huyền ) × (cosin góc kề)

Ví dụ:

Trong tam giác ABC vuông tại A, ta có: \(\begin{array}{l}b = a.\sin B = a.\cos C;\\c = a.\sin C = a.\cos B.\end{array}\)

4. Cách tính cạnh góc vuông theo cạnh góc vuông còn lại và tỉ số lượng giác của góc nhọn

Trong tam giác vuông, mỗi cạnh góc vuông bằng cạnh góc vuông kia nhân với tang góc đối hoặc côtang góc kề.

Cạnh góc vuông = (cạnh góc vuông còn lại ) × (tan góc đối) 

= (cạnh góc vuông còn lại ) × (cot góc kề)

Ví dụ:

Trong tam giác ABC vuông tại A, ta có: \(\begin{array}{l}b = c.\tan B = c.\cot C;\\c = b.\tan C = b.\cot B.\end{array}\)

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