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Giải bài 5 trang 24 Chuyên đề học tập Toán 11 Cánh diều

Cho hai đường tròn (O1; R) và (O2; R) tiếp xúc ngoài với nhau tại A (Hình 39).

Tổng hợp đề thi ♓học kì 2 lớp 11 tất cả các môn - Cánh diều

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Đề bài

Cho hai đường tròn (O1; R) và (O2; R) tiếp xúc ngoài với nhau tại A (Hình 39).

a) Tìm phép tịnh tiến biến đường tròn (O1) thành đường tròn (O2).

b) Tìm phép đối xứng tâm biến đường tròn (O1) thành đường tròn (O2).

c) Tìm phép đối xứng trục biến đường tròn (O1) thành đường tròn (O2).

Phương pháp giải - Xem chi tiết

Dựa vào kiến thức: - Cho vectơ \(\overrightarrow u \), phép tịnh tiến theo vectơ \(\overrightarrow u \) là phép biến hình biến điểm M thành  điểm M’ sao cho \(\overrightarrow {MM'}  = \overrightarrow u \). - Cho điểm O, phép biến hình biến điểm O thành chính nó và biến mỗi điểm \(M \ne O\) thành điểm M’ sao cho O  là trung điểm của MM’ được gọi là phép đối xứn tâm O, kí hiệu \({Đ_O}\). Điểm O được gọi là tâm đối xứng. - Cho đường thẳng d. Phép biến hình biến mỗi điểm M thuộc d thành chính nó, biến mỗi điểm M không thuộc d thành điểm M' sao cho d là đường trung trực của đoạn thẳng MM' được gọi là phép đối xứng trục d. Kí hiệu \({Đ_d}\). 

Lời giải chi tiết

a) Hai đường tròn \(({O_1};{\rm{ }}R)\) và \(({O_2};{\rm{ }}R)\) có cùng bán kính. Ta có phép tịnh tiến theo vectơ \(\overrightarrow {{O_1}{O_2}} \) biến điểm tâm \({O_1}\) thành tâm \({O_2}\).Như vậy, phép tịnh tiến theo vectơ \(\overrightarrow {{O_1}{O_2}} \) biến đường tròn \(({O_1};{\rm{ }}R)\)thành đường tròn \(({O_2};{\rm{ }}R)\)

b) Ta có: \({O_1}A{\rm{ }} = {\rm{ }}{O_2}A{\rm{ }} = {\rm{ }}R\) nên A là trung điểm của \({O_1}{O_2}\). Do đó, có phép đối xứng tâm A biến O1 thành O2.

Như vậy, phép đối xứng tâm O biến đường tròn \(({O_1};{\rm{ }}R)\) thành đường tròn \(({O_2};{\rm{ }}R)\).c)

 

Qua A, kẻ đường thẳng d vuông góc với \({O_1}{O_2}.\)Khi đó đường thẳng d là đường trung trực của đoạn thẳng O1O2. Do đó, ta có phép đối xứng trục d biến O1 thành O2.

Như vậy, phép đối xứng trục d biến đường tròn \(({O_1};{\rm{ }}R)\) thành đường tròn \(({O_2};{\rm{ }}R)\).

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