Question

3．直角△ABC的三個(gè)頂點(diǎn)都在單位圓x²+y²=1上，點(diǎn)M（$\frac{1}{2}$，$\frac{1}{2}$）．則|$\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}$|最大值是（　�。�

• A． $\sqrt{2}+1$
• B． $\sqrt{2}+2$
• C． $\frac{3\sqrt{2}}{2}+1$
• D． $\frac{3\sqrt{2}}{2}+2$

Answer 1

分析 由題意，|$\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}$|=|$\overrightarrow{MA}$+2$\overrightarrow{MO}$|≤|$\overrightarrow{MA}$|+2|$\overrightarrow{MO}$|，當(dāng)且僅當(dāng)M，O，A共線同向時(shí)，取等號(hào)，即可求出|$\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}$|的最大值．

解答 解：由題意，|$\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}$|=|$\overrightarrow{MA}$+2$\overrightarrow{MO}$|≤|$\overrightarrow{MA}$|+2|$\overrightarrow{MO}$|，
當(dāng)且僅當(dāng)M，O，A共線同向時(shí)，取等號(hào)，即|$\overrightarrow{MA}+\overrightarrow{MB}+\overrightarrow{MC}$|取得最大值，最大值是$\sqrt{2}$+$\frac{\sqrt{2}}{2}$+1=$\frac{3\sqrt{2}}{2}$+1，
故選：C．

點(diǎn)評(píng) 本題考查點(diǎn)與圓的位置關(guān)系，考查向量知識(shí)的運(yùn)用，比較基礎(chǔ)．