Question

已知平行四邊形中,三個頂點(diǎn)對應(yīng)的復(fù)數(shù)分別是2+i,4+3i,3+5i,求第四個頂點(diǎn)對應(yīng)的復(fù)數(shù).

Answer 1

圖3-2-1

解:如圖3-2-1,設(shè)點(diǎn)Z₁,Z₂,Z₃分別對應(yīng)復(fù)數(shù)2+i,4+3i,3+5i.?

(1)若Z₁Z₃為對角線,則=,?

即z₄-z₁=z₃-z₂,∴z₄=z₃-z₂+z₁=(3+5i)-(4+5i)+(2+i)=1+3i.?

(2)若Z₁Z₂為對角線,則=,?

即z₄-z₁=z₂-z₃,∴z₄=z₂-z₃+z₁=(4+3i)-(3+5i)+(2+i)=3-i.?

(3)若Z₂Z₃為對角線,則=,即z₄-z₂=z₃-z₁,?

∴z₄=z₃-z₁+z₂=(3+5i)-(2+i)+(4+3i)=5+7i.

點(diǎn)評:本題中平行四邊形共三種情況,故第四點(diǎn)對應(yīng)的復(fù)數(shù)應(yīng)有三解,要防止發(fā)生以偏概全的錯誤.