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歡迎您訪問數學考題練習:如圖,在△ABC中,∠BAC=120°,P是△!

數學考題練習:如圖,在△ABC中,∠BAC=120°,P是△

更新時間:2024-01-12 16:33:15作者:貝語網校

如圖,在△ABC中,∠BAC=120°,P是△ABC內一點,則

A.PA+PB+PC<AB+AC

B.PA+PB+PC>AB+AC

C.PA+PB+PC=AB+AC

D.PA+PB+PC與AB+AC的大小關系不確定,與P點位置有關

試題答案

B

試題解析

把△APC繞A逆時針旋轉60°得到△AP′C′,根據旋轉的性質得到∠CAC′=∠PAP′=60°,AC=AC′,AP=AP′,PC=P′C,則有△APP′為等邊三角形,得PP′=AP;又∠BAC=120°,得到B,A,C′共線,根據兩點之間線段最短得到BC′<BP+PP′+P′C,即得到AB+AC<AP+BP+CP.

解答:解:把△APC繞A逆時針旋轉60°得到△AP′C′,如圖

∴∠CAC′=∠PAP′=60°,AC=AC′,AP=AP′,PC=P′C,

∴△APP′為等邊三角形,

∴PP′=AP,

∵∠BAC=120°,

∴∠BAC′=120°+60°=180°,

即B,A,C′共線,

∴BC′<BP+PP′+P′C,

即AB+AC<AP+BP+CP.

故選B.

點評:本題考查了旋轉的性質:旋轉前后的兩個圖形全等,對應點與旋轉中心的連線段的夾角等于旋轉角,對應點到旋轉中心的距離相等.同時考查了兩點之間線段最短.

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