92 學測數學第 4 題

Question

設 $ABC$ 為坐標平面上一三角形，$P$ 為平面上一點且 $\overset{\largeightharpoonup}{AP}=\dfrac{1}{5}\overset{\largeightharpoonup}{AB}+\dfrac{2}{5}\overset{\largeightharpoonup}{AC}$，則 $\dfrac{	riangle ABP 	ext{面積}}{	riangle ABC 	ext{面積}}$ 等於

Accepted Answer

答案：$(3)$
令 $\overset{\large\rightharpoonup}{AB}=u$、$\overset{\large\rightharpoonup}{AC}=v$，則 $\overset{\large\rightharpoonup}{AP}=\dfrac{1}{5}u+\dfrac{2}{5}v$。面積比為 $\dfrac{|\det(u,\overset{\large\rightharpoonup}{AP})|}{|\det(u,v)|}=\dfrac{|\det(u,\frac{1}{5}u+\frac{2}{5}v)|}{|\det(u,v)|}=\dfrac{2}{5}$。故選 $(3…