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Giải bài tập 7.11 trang 38 SGK Toán 9 tập 2 - Cùng khám phá

Cho ABCD là tứ giác nội tiếp. a) Chứng minh rằng \(\widehat {BAC} = \widehat {BDC}\). b) AC cắt BD tại M. Chứng minh rằng MA.MC = MB.MD.

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Đề bài

Cho ABCD là tứ giác nội tiếp.

a) Chứ꧙ng minh rằng \(\wideha𓆏t {BAC} = \widehat {BDC}\).

b) AC cắt BD tại M. Chứng minh rằng MA❀.MC = MB.MD.

Phương pháp giải - Xem chi tiết

Đọc kĩ dữ kiện để vẽ hình Áp dụng hai góc nội tiếp cùng chắn một cung thì bằng nhau. Chứng minh \(\Delta \)BAM đồng dạng \(\Delta \)CDM suy ra tỉ lệ các cạnh bằng nhau.

Lời giải chi tiết

a) Ta có \(\widehat {BAC}\) và \(\widehat {BDC}\) là hai góc nội tiếp cùng chắn cung BC nhỏ nên \(\widehat {BAC} = \widehat {BDC}\). b) Xét \(\Delta \)BAM và \(\Delta \)CDM có \(\widehat {BAM} = \widehat {MDC}\) (cmt) \(\widehat {AMB} = \widehat {CMD}\) (đối đỉnh) Suy ra \(\Delta \)BAM \(\backsim \) \(\Delta \)CDM (g-g) Nên \(\frac{{MA}}{{MB}} = \frac{{MD}}{{MC}}\) hay MA.MC = MB.MD.

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