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Giải bài 9.35 trang 109 SGK Toán 8 tập 2 - Kết nối tri thức

Cho tam giác ABC vuông tại A có đường cao AH.

ꦺ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 ABC vuông tại A có đường cao AH. Cho M và N lần lượt là trung điểm của AB và AC. Chứng minh ΔHBM ∽ ΔHAN 

Video hướng dẫn giải

Phương pháp giải - Xem chi tiết

Chứng minh \(\widehat {HAC} = \widehat {ABC} = \widehat {ABH}\) và \(\frac{{HB}}{{HA}} = \frac{{BM}}{{AN}}\) suy ra ΔHBM ∽ ΔHAN 

Lời giải chi tiết

Xét \(\Delta BAC\) và \(\Delta BHA\) có:

\(\widehat A = \widehat H\)

\(\widehat B\) chung

🦋nên \(\Delta BAC \backsim \Delta BHA\left( g.g \right)\)

suy ra \(\frac{{BA}}{{BH}} = \frac{{AC}}{{HA}} \)

🎐Do đó \(\frac{{HB}}{{HA}} = \frac{{BA}}{{AC}}(1)\)

Xét \(\Delta BAC\) và \(\Delta AHC\) có:

\(\widehat A = \widehat H\)

\(\widehat C\) chung

𒐪nên \(\Delta BAC \backsim \Delta AHC\left( g.g \right)\)

suy ra \(\widehat {HAC} = \widehat {ABC} (2)\)

ౠVì M là trung điểm của AB nên \(\frac{{BM}}{{BA}} = \frac{1}{2}\)

ꦜVì N là trung điểm của AC nên \(\frac{{AN}}{{AC}} = \frac{1}{2}\)

Suy ra \(\frac{{BM}}{{BA}} = \frac{{AN}}{{AC}}\)

🦩do đó \(\frac{{BM}}{{AN}} = \frac{{BA}}{{AC}}(3)\)

𝕴Từ (1), (3) suy ra: \(\frac{{HB}}{{HA}} = \frac{{BM}}{{AN}}\)

Xét hai tam giác HBM và HAN có:

🐼\(\widehat {HAC} = \widehat {ABC} = \widehat {ABH}\)

\(\frac{{HB}}{{HA}} = \frac{{BM}}{{AN}}\)

suy ra \(\Delta HBM \backsim \Delta HAN\) (c.g.c)

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Tham Gia Group Dành Cho Lớp 8 Chia Sẻ, Trao Đổi Tài Liệu Miễn Phí

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