已知正三棱台上底边为3,下底边为6,高为1,求斜高与侧棱长 一个正三棱台的上 下底面边长分别为3和6,高是2(1)求斜高...
\u5df2\u77e5\u6b63\u4e09\u68f1\u53f0ABC\uff0dA'B'C'\u7684\u9ad8\u4e3a\u4e00\uff0c\u4e0a\u5e95\u8fb9\u957f\u4e3a\u4e8c\uff0c\u4e0b\u5e95\u8fb9\u957f\u4e3a\u56db\uff0c\u6c42\u5b83\u5e95\u4fa7\u68f1\u957f\u4e2a\u659c\u9ad8\u3002\u505a\u25b3ABC\u548c\u25b3A'B'C'\u8fb9\u4e0a\u7684\u9ad8CF\u3001BE\u3001C'F'\u3001B'E'\u76f8\u4ea4\u4e8e\u70b9O\u3001O'\u3002
\u8fde\u63a5OO'\u3001EE'\uff0c\u505aEP\u3001CQ\u2225OO'\u3002
\u2235\u4e09\u68f1\u53f0ABC\uff0dA'B'C'\u662f\u6b63\u4e09\u68f1\u53f0
\u2234OO'\u5782\u76f4\u25b3ABC\u548c\u25b3A'B'C'\uff1b\u5373EP\u3001CQ\u4e5f\u5782\u76f4\u25b3ABC\u548c\u25b3A'B'C'\u3002
\u2234\u4fa7\u68f1\u957fCC'=\u221a(CQ^2+C'Q^2)\uff0c\u4fa7\u9762\u659c\u9ad8EE'=\u221a(EP^2+E'P^2)\u3002
\u53c8\u2235\u25b3ABC\u548c\u25b3A'B'C'\u662f\u6b63\u4e09\u89d2\u5f62\uff0cCF\u3001BE\u3001C'F'\u3001B'E'\u662f\u5176\u8fb9\u4e0a\u7684\u9ad8\uff0cO\u3001O'\u4e3a\u4ea4\u70b9
\u2234OE=(1/3)BE\uff0cO'E'=(1/3)B'E'\uff0cOB=(2/3)BE\uff0cO'B'=(2/3)B'E'\u3002
\u53c8\u2235BE=sin60BC=\u221a3\uff0cBE=sin60BC=2\u221a3
\u2234OE=(1/3)BC=(\u221a3)/3\uff0cO'E'=(1/3)B'C'=(2\u221a3)/3\uff0c
OC=OB=(2/3)BE=(2\u221a3)/3\uff0cO'C'=O'B'=(2/3)B'E'=(4\u221a3)/3
\u2234PE'=O'E'-OE=(\u221a3)/3\uff0cQC'=O'C'-OC=(2\u221a3)/3
\u2234\u4fa7\u68f1\u957fCC'=\u221a(CQ^2+C'Q^2)=\u221a{1^2+[(2\u221a3)/3]^2}=(\u221a21)/3\uff0c
\u4fa7\u9762\u659c\u9ad8EE'=\u221a(EP^2+E'P^2)=\u221a{1^2+[(\u221a3)/3]^2}=(2\u221a3)/3\u3002
\u7b54\uff1a
\u5982\u4e0b\u56fe\uff0c\u6b63\u4e09\u89d2\u5f62ABC\u8fb9\u957f\u4e3a6\uff0c\u6b63\u4e09\u89d2\u5f62DEF\u8fb9\u957f\u4e3a3
\u6b63\u4e09\u68f1\u53f0\u9ad8h=2\uff0c\u8bbe\u6b63\u4e09\u68f1\u9525O-ABC\u7684\u9ad8\u4e3aH\uff0c\u5219\u6b63\u4e09\u68f1\u9525O-DEF\u7684\u9ad8\u4e3aH-h
\u6839\u636e\u76f8\u4f3c\u6027\u8d28\u6709\uff1a
(H-h)/H=DE/AB=3/6=1/2
(H-2)/H=1/2
\u89e3\u5f97\uff1aH=4
\u6b63\u4e09\u89d2\u5f62ABC\u5e95\u8fb9\u4e0a\u7684\u9ad8=6sin60\u00b0=3\u221a3
\u9876\u70b9C\u5230\u5e95\u9762\u4e2d\u5fc3\u7684\u8ddd\u79bb=2\u221a3
\u6839\u636e\u52fe\u80a1\u5b9a\u7406\uff1aCO^2=(2\u221a3)^2+H^2=12+16=28
\u89e3\u5f97\uff1aCO=2\u221a7
\u6839\u636e\u76f8\u4f3c\u6c42\u5f97\uff1aOF/OC=1/2
\u6240\u4ee5\uff1aOF=FC=\u221a7
\u603b\u659c\u9ad8L^2=H^2+(\u221a3)^2=16+3=19
L=\u221a19
\u6240\u4ee5\uff1a\u659c\u9ad8=\u221a19 /2
\u4fa7\u9762\u4e3a\u4e09\u4e2a\u7b49\u8170\u68af\u5f62\uff1a
S=3*(3+6)*\u221a(19/2) /2=27\u221a19 /4
\u8868\u9762\u79ef\u518d\u52a0\u4e0a\u4e0a\u4e0b\u5e95\u9762\u7684\u9762\u79ef\uff1a
S=27\u221a19/4+(9/2)*(\u221a3/2)+(36/2)*(\u221a3/2)
=27\u221a19/4+45\u221a3/4
令C,D为同一侧面上上下底边的中点,
过BC做底面的垂线,垂足分别为EF,则E,F均在AD上,
∵正三棱台上底边为3,下底边为6,高为1,
∴OC=
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