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Giải bài 4 trang 79 SGK Toán 7 tập 2 - Cánh diều

Cho tam giác ABC và điểm M thuộc cạnh BC thỏa mãn

✅Tổng hợp đề thi học kì 2 lớp 7 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên...
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Đề bài

Cho tam giác ABC và điểm M thuộc cạnh BC♕ thỏa mãn \(\Delta AMB = \Delta AMC\) (Hình 32). Chứng minh rằng:

 

a) M là trung điểm của đoạn thẳng BC.

b) Tia AM là tia phân giác của góc BAC và \(AM \bot BC\).

Phương pháp giải - Xem chi tiết

Hai tam giác bằng nhau là hai tam giác có các cạnh tương ứng và các góc tương ứng bằng nhau.

a) Muốn chứng minh M là trung điểm của đoạn thẳng BC, ta chứng minh MB = MC.

b) Muốn chứng minh tia AM là phân giác của góc BAC ta chứng minh góc BAM = góc CAM.

Trong một tam giác, một đường thẳng vừa là trung tuyến vừa là phân giác thì đường thẳng đó vuông góc với đáy tương ứng. Hoặc ta có thể chứng minh góc được tạo bởi hai đường thẳng đó có số đo góc là 90°.

Lời giải chi tiết

a) Ta có: \(\Delta AMB = \Delta AMC\) nên AB = AC, MB = MC nên M là trung điểm của đoạn thẳng BC.

b) Ta có: \(\Delta AMB = \Delta AMC\) nên \(\widehat {AMB} = \widehat {AMC},\widehat {MAB} = \widehat {MAC},\widehat {MBA} = \widehat {MCA}\).

Vậy tia AM là tia phân giác của góc BAC vì \(\widehat {MAB} = \widehat {MAC}\).

Ta thấy: \(\widehat {AMB} = \widehat {AMC}\) mà ba điểm B, M, C thẳng hàng nên \(\widehat {BMC} = 180^\circ \).

Suy ra \(\widehat {AMB} = \widehat {AMC} = \dfrac{1}{2}.\widehat {BMC} = \dfrac{1}{2}.180^\circ  = 90^\circ \). Vậy \(AM \bot BC\).

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