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Giải mục 2 trang 53, 54, 55 SGK Toán 8 – Cánh diều

Quan sát Hình 3 và cho biết:

🅷Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Cánh diều

Toán - Văn - Anh - Khoa học tự nhiên
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Lựa chọn câu để xem lời giải nhanh hơn

HĐ2

Video hướng dẫn giải

Quan sát Hình 3 và cho biết:

a)      Đường thẳng \(d\) có song song với BC hay không? b)     Bằng cách đếm số ô vuông, dự đoán xem các tỉ số \(\frac{{AM}}{{MB}},\frac{{AN}}{{NC}}\) có bằng nhau hay không?

 

Phương pháp giải:

Quan sát hình và trả lời câu hỏi.

Lời giải chi tiết:

a)      Quan sát hình ta thấy \(d\parallel BC\). b)     Ta thấy: Độ dài AM là 2 lần cạnh của một ô vuông. Độ dài MB là cạnh của một ô vuông. \( \Rightarrow \frac{{AM}}{{MB}} = \frac{2}{1} = 2\) Độ dài AN là 2 lần đường chéo của một ô vuông. Độ dài NC là độ dài đường chéo của một ô vuông. \( \Rightarrow \frac{{AN}}{{NC}} = \frac{2}{1} = 2\) Vậy \(\frac{{AM}}{{MB}} = \frac{{AN}}{{NC}}\).

LT1

Video hướng dẫn giải

Trong Hình 4🗹, chứng tỏ rằng nếu \(MN\parallel BC\) thì \(\frac{{MB}}{{AB}} = \frac{{NC}}{{AC}}\).

Phương pháp giải:

Dựa vào định lý Thales để chứng minh hai tỉ số bằng nhau.

Lời giải chi tiết:

Xét tam giác ABC với \(MN\parallel BC\), ta có \(\frac{{MB}}{{AB}} = \frac{{NC}}{{AC}}\) (định lý Thales).

LT2

Video hướng dẫn giải

Cho tam giác ABC có G là trọng tâm. Đường thẳng qua G song song với BC lần lượt cắt AB, AC tại M, N. Chứng minh \( \frac{AM}{AB} = \frac{AN}{AC} = \frac{2}{3} \).

Phương pháp giải:

Sử dụng định lý Thales để chứng minh \( \frac{AM}{AB} = \frac{AN}{AC} = \frac{2}{3} \).

Lời giải chi tiết:

Gọi AD là đường trung tuyến của tam giác ABC (D \(\in\) BC) Vì G là trọng tâm của tam giác ABC nên AG = \(\frac{2}{3}\) AD hay \(\frac{AG}{AD} =\frac{2}{3}\) . Xét tam giác ABD với MG // BD, ta có:  \( \frac {AM}{AB} = \frac{AG}{AD} =\frac{2}{3}\) (Định lí Thales) (1) Tương tự, xét  tam giác ADC với GN // DC, ta có:  \( \frac {AN}{AC} = \frac{AG}{AD} =\frac{2}{3}\) (Định lí Thales) (2) Từ (1) và (2) suy ra \( \frac{AM}{AB} = \frac{AN}{AC} = \frac{2}{3} \) (đpcm).

HĐ3

Video hướng dẫn giải

Trong Hình 7, cho AM = 1, MB = 2, AN = 1,5, NC = 3.

a)      So sánh các tỉ số \(\frac{{AM}}{{MB}};\,\,\frac{{AN}}{{NC}}\). b)     Đường thẳng \(d\) (đi qua M, N) có song song với BC hay không?

 

Phương pháp giải:

a)      Dựa vào số liệu đã cho, tính và so sánh các tỉ số. b)     Quan sát hình vẽ và cho biết đường thẳng \(d\) (đi qua M, N) có song song với BC hay không.

Lời giải chi tiết:

a) \(\frac{{AM}}{{MB}} = \frac{1}{2}\) \(\frac{{AN}}{{AC}} = \frac{{1,5}}{3} = \frac{1}{2}\) Vậy \(\frac{{AM}}{{MB}} = \frac{{AN}}{{NC}}\). b) Qua B kẻ đường thẳng song song với đường thẳng d, cắt AC tại C’.

Xét ∆ABC’ với MN // BC’, ta có: \( \frac{AM}{MB}=\frac{AN}{NC′}\) (định lí Thalès). Mà theo câu a, \(\frac{{AM}}{{MB}} = \frac{{AN}}{{NC}}\) nên ta có \(\frac{{AN}}{{NC}} = \frac{AN}{NC′}\) Suy ra NC = NC’ hay C và C’ là hai điểm trùng nhau. Do đó C nằm trên đường thẳng đi qua B và song song với đường thẳng d. Vậy đường thẳng d (đi qua M, N) song song với BC.

LT3

Video hướng dẫn giải

Cho tam giác ABC vuông tại A có CA = 4, CB = 5. Giả sử M, N là hai điểm lần lượt nằm trên hai cạnh CA, CB sao cho CM = 1, CN = 1,25. Tính độ dài đoạn thẳng MN.

Phương pháp giải:

-         Sử dụng định lý Thales đảo để chứng minh \(MN\parallel AB\). -         Chứng minh \(MN \bot AC\) -         Sử dụng định lý Pytago để tính độ dài cạnh MN.

Lời giải chi tiết:

Xét tam giác ABC có \(\begin{array}{l}\frac{{CM}}{{CA}} = \frac{1}{4}\\\frac{{CN}}{{CB}} = \frac{{1,25}}{5} = \frac{1}{4}\\ \Rightarrow \frac{{CM}}{{CA}} = \frac{{CN}}{{CB}}\end{array}\) \( \Rightarrow MN\parallel AB\) (Định lý Thales đảo) Mà \(AB \bot AC\) nên \(MN \bot AC\) hay tam giác MNC vuông tại M Xét tam giác MNC vuông tại M có: \(MC = 1,\,\,NC = 1,25\). Theo định lý Pytago ta có: \(\begin{array}{l}M{N^2} + M{C^2} = N{C^2}\\\,\,\,\,\,\,\,M{N^2} + {1^2} = 1,{25^2}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,M{N^2} = 1,{25^2} - {1^2}\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,M{N^2} = 0,5625\\\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,MN = 0,75\end{array}\) Vậy MN = 0,75.

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