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Giải bài 5 trang 98 vở thực hành Toán 8 tập 2

Cho tam giác A'B'C' đồng dạng với tam giác ABC theo tỉ số k. Gọi A'H' và AH lần lượt là các đường cao đỉnh A' và A của tam giác A'B'C' và tam giác ABC.

💝Tổng hợp đề thi học kì 2 lớp 8 tất cả các môn - Kết nối tri thức

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

Cho tam giác A'B'C' đồng dạng với tam giác ABC theo tỉ số k. Gọi A'H' và AH lần lượt là các đường cao đỉnh A' và A của tam giác A'B'C' và tam giác ABC. a) Chứng minh rằng \(\frac{{A}'{H}'}{AH}=k\);

b) Diện tích tam giác A'B'C' bằng k2 lần diện tích tam giác ABC.

Phương pháp giải - Xem chi tiết

a) Vì ΔA′B′C′ ∽ ΔABC 

=> \(\hat{B}=\widehat{{{B}'}};\frac{{A}'{H}'}{AB}=\frac{{A}'{C}'}{AC}=\frac{{B}'{C}'}{BC}=k\)

🎃Chứng minh ΔA′H′B′ ∽ ΔAHB  suy ra các hệ số tỉ lệ và chứng minh được \(\frac{{A}'{H}'}{AH}=k\).

b) Tính diện tích tam giác ABC và A”B”C” từ đó sẽ xét tỉ số diện tích của hai tam giác đó.

Lời giải chi tiết

a) Hai tam giác vuông AHB (vuông tại H) và A’H’B’ (vuông tại H’) có $\widehat{ABH}=\widehat{A'B'H'}$.Do đó $\Delta AHB\backsim \Delta A'H'B'$ (một cặp góc nhọn bằng nhau).Suy ra $\frac{A'H'}{AH}=\frac{A'B'}{AB}=k$.

b) Vì $\frac{A'H'.B'C'}{2}={{k}^{2}}.\frac{AH.BC}{2}$ nên diện tích tam giác A’B’C’ bằng k2 lần diện tích tam giác ABC.

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