初三数学题目。 如图①所示,在直角梯形ABCD中,∠BAD90°,E是直线AB上一点,过E作直线L‖

\u5982\u56fe\u2460\u6240\u793a\uff0c\u5728\u76f4\u89d2\u68af\u5f62ABCD\u4e2d\uff0c\u2220BAD=90\u00b0\uff0cE\u662f\u76f4\u7ebfAB\u4e0a\u4e00\u70b9\uff0c\u8fc7E\u4f5c\u76f4\u7ebfl\u2225BC\uff0c\u4ea4\u76f4\u7ebfCD\u4e8e\u70b9F\uff0e\u5c06\u76f4\u7ebfl

\u7531\u9898\u610f\u5f97\uff1a\uff081\uff09AB=2\uff0e\uff082\uff09S\u68af\u5f62ABCD=12\uff0e\uff083\uff09\u5f53\u5e73\u79fb\u8ddd\u79bbBE\u5927\u4e8e\u7b49\u4e8e4\u65f6\uff0c\u76f4\u89d2\u68af\u5f62ABCD\u88ab\u76f4\u7ebfl\u626b\u8fc7\u7684\u9762\u79ef\u6052\u4e3a12\uff0e\uff084\uff09\u5f532\uff1ct\uff1c4\u65f6\uff0c\u5982\u56fe\u6240\u793a\uff0c\u76f4\u89d2\u68af\u5f62ABCD\u88ab\u76f4\u7ebfl\u626b\u8fc7\u7684\u9762\u79efS=S\u76f4\u89d2\u68af\u5f62ABCD-SRt\u25b3DOF=12-12\uff084-t\uff09\u00d72\uff084-t\uff09=-t2+8t-4\uff0e\uff085\uff09\u2460\u5f530\uff1ct\uff1c2\u65f6\uff0c\u67094t\uff1a\uff0812-4t\uff09=1\uff1a3\uff0c\u89e3\u5f97t=34\uff0e\u2461\u5f532\uff1ct\uff1c4\u65f6\uff0c\u6709\uff08-t2+8t-4\uff09\uff1a[12-\uff08-t2+8t-4\uff09]=3\uff1a1\uff0c\u5373t2-8t+13=0\uff0c\u89e3\u5f97t=4-3\uff0ct=4+3 \uff08\u820d\u53bb\uff09\uff0e\u7b54\uff1a\u5f53t=34 \u6216t=4-3\u65f6\uff0c\u76f4\u7ebfl\u5c06\u76f4\u89d2\u68af\u5f62ABCD\u5206\u6210\u7684\u4e24\u90e8\u5206\u9762\u79ef\u4e4b\u6bd4\u4e3a1\uff1a3\uff0e

\u5e0c\u671b \u5bf9\u4f60 \u6709\u5e2e\u52a9
\u5e0c\u671b\u91c7\u7eb3 12999 \u9898\u8c37
\uff082012\u91cd\u5e86\u5e0212\u5206\uff09\u5df2\u77e5\uff1a\u5982\u56fe\uff0c\u5728\u76f4\u89d2\u68af\u5f62ABCD\u4e2d\uff0cAD\u2225BC\uff0c\u2220B=90\u00b0\uff0cAD=2\uff0cBC=6\uff0cAB=3\uff0eE\u4e3aBC\u8fb9\u4e0a\u4e00\u70b9\uff0c\u4ee5BE\u4e3a\u8fb9\u4f5c\u6b63\u65b9\u5f62BEFG\uff0c\u4f7f\u6b63\u65b9\u5f62BEFG\u548c\u68af\u5f62ABCD\u5728BC\u7684\u540c\u4fa7\uff0e
\uff081\uff09\u5f53\u6b63\u65b9\u5f62\u7684\u9876\u70b9F\u6070\u597d\u843d\u5728\u5bf9\u89d2\u7ebfAC\u4e0a\u65f6\uff0c\u6c42BE\u7684\u957f\uff1b
\uff082\uff09\u5c06\uff081\uff09\u95ee\u4e2d\u7684\u6b63\u65b9\u5f62BEFG\u6cbfBC\u5411\u53f3\u5e73\u79fb\uff0c\u8bb0\u5e73\u79fb\u4e2d\u7684\u6b63\u65b9\u5f62BEFC\u4e3a\u6b63\u65b9\u5f62B\u2032EFG\uff0c\u5f53\u70b9E\u4e0e\u70b9C\u91cd\u5408\u65f6\u505c\u6b62\u5e73\u79fb\uff0e\u8bbe\u5e73\u79fb\u7684\u8ddd\u79bb\u4e3at\uff0c\u6b63\u65b9\u5f62B\u2032EFG\u7684\u8fb9EF\u4e0eAC\u4ea4\u4e8e\u70b9M\uff0c\u8fde\u63a5B\u2032D\uff0cB\u2032M\uff0cDM\uff0c\u662f\u5426\u5b58\u5728\u8fd9\u6837\u7684t\uff0c\u4f7f\u25b3B\u2032DM\u662f\u76f4\u89d2\u4e09\u89d2\u5f62\uff1f\u82e5\u5b58\u5728\uff0c\u6c42\u51fat\u7684\u503c\uff1b\u82e5\u4e0d\u5b58\u5728\uff0c\u8bf7\u8bf4\u660e\u7406\u7531\uff1b
\uff083\uff09\u5728\uff082\uff09\u95ee\u7684\u5e73\u79fb\u8fc7\u7a0b\u4e2d\uff0c\u8bbe\u6b63\u65b9\u5f62B\u2032EFG\u4e0e\u25b3ADC\u91cd\u53e0\u90e8\u5206\u7684\u9762\u79ef\u4e3aS\uff0c\u8bf7\u76f4\u63a5\u5199\u51faS\u4e0et\u4e4b\u95f4\u7684\u51fd\u6570\u5173\u7cfb\u5f0f\u4ee5\u53ca\u81ea\u53d8\u91cft\u7684\u53d6\u503c\u8303\u56f4\uff0e

\u3010\u7b54\u6848\u3011\u89e3\uff1a\uff081\uff09\u5982\u56fe\u2460\uff0c\u8bbe\u6b63\u65b9\u5f62BEFG\u7684\u8fb9\u957f\u4e3ax\uff0c
\u5219BE=FG=BG=x\u3002
\u2235AB=3\uff0cBC=6\uff0c\u2234AG=AB\ufe63BG=3\ufe63x\u3002
\u2235GF\u2225BE\uff0c\u2234\u25b3AGF\u223d\u25b3ABC\u3002
\u2234 \uff0c\u5373 \u3002
\u89e3\u5f97\uff1ax=2\uff0c\u5373BE=2\u3002
\uff082\uff09\u5b58\u5728\u6ee1\u8db3\u6761\u4ef6\u7684t\uff0c\u7406\u7531\u5982\u4e0b\uff1a
\u5982\u56fe\u2461\uff0c\u8fc7\u70b9D\u4f5cDH\u22a5BC\u4e8eH\uff0c
\u5219BH=AD=2\uff0cDH=AB=3\uff0c
\u7531\u9898\u610f\u5f97\uff1aBB\u2032=HE=t\uff0cHB\u2032=|t\ufe632|\uff0cEC=4\ufe63t\uff0c
\u2235EF\u2225AB\uff0c\u2234\u25b3MEC\u223d\u25b3ABC\u3002
\u2234 \uff0c\u5373 \u3002\u2234ME=2\ufe63 t\u3002
\u5728Rt\u25b3B\u2032ME\u4e2d\uff0cB\u2032M2=ME2+B\u2032E2=22+\uff082\ufe63 t\uff092= t2\ufe632t+8\u3002
\u5728Rt\u25b3DHB\u2032\u4e2d\uff0cB\u2032D2=DH2+B\u2032H2=32+\uff08t\ufe632\uff092=t2\ufe634t+13\u3002
\u8fc7\u70b9M\u4f5cMN\u22a5DH\u4e8eN\uff0c\u5219MN=HE=t\uff0cNH=ME=2\ufe63 t\uff0c
\u2234DN=DH\ufe63NH=3\ufe63\uff082\ufe63 t\uff09= t+1\u3002
\u5728Rt\u25b3DMN\u4e2d\uff0cDM2=DN2+MN2=\uff08 t+1\uff092+ t 2= t2+t+1\u3002
\uff08\u2160\uff09\u82e5\u2220DB\u2032M=90\u00b0\uff0c\u5219DM2=B\u2032M2+B\u2032D2\uff0c
\u5373 t2+t+1=\uff08 t2\ufe632t+8\uff09+\uff08t2\ufe634t+13\uff09\uff0c\u89e3\u5f97\uff1at= \u3002
\uff08\u2161\uff09\u82e5\u2220B\u2032MD=90\u00b0\uff0c\u5219B\u2032D2=B\u2032M2+DM2\uff0c
\u5373t2\ufe634t+13=\uff08 t2\ufe632t+8\uff09+\uff08 t2+t+1\uff09\uff0c\u89e3\u5f97\uff1at1=\ufe633+ \uff0ct2=\ufe633\ufe63 \uff08\u820d\u53bb\uff09\u3002
\u2234t=\ufe633+ \u3002
\uff08\u2162\uff09\u82e5\u2220B\u2032DM=90\u00b0\uff0c\u5219B\u2032M2=B\u2032D2+DM2\uff0c
\u5373 t2\ufe632t+8=\uff08t2\ufe634t+13\uff09+\uff08 t2+t+1\uff09\uff0c\u6b64\u65b9\u7a0b\u65e0\u89e3\u3002
\u7efc\u4e0a\u6240\u8ff0\uff0c\u5f53t= \u6216\ufe633+ \u65f6\uff0c\u25b3B\u2032DM\u662f\u76f4\u89d2\u4e09\u89d2\u5f62\uff1b
\uff083\uff09 \u3002
\uff082012\u9ed1\u9f99\u6c5f\u7ee5\u53168\u5206\uff09\u5982\u56fe\uff0c\u70b9E\u662f\u77e9\u5f62ABCD\u7684\u5bf9\u89d2\u7ebfBD\u4e0a\u7684\u4e00\u70b9\uff0c\u4e14BE=BC\uff0cAB=3\uff0cBC=4\uff0c\u70b9P\u4e3a\u76f4\u7ebfEC\u4e0a\u7684\u4e00\u70b9\uff0c\u4e14PQ\u22a5BC\u4e8e\u70b9Q\uff0cPR\u22a5BD\u4e8e\u70b9R\uff0e
\uff081\uff09\u5982\u56fe1\uff0c\u5f53\u70b9P\u4e3a\u7ebf\u6bb5EC\u4e2d\u70b9\u65f6\uff0c\u6613\u8bc1\uff1aPR+PQ= \uff08\u4e0d\u9700\u8bc1\u660e\uff09\uff0e
\uff082\uff09\u5982\u56fe2\uff0c\u5f53\u70b9P\u4e3a\u7ebf\u6bb5EC\u4e0a\u7684\u4efb\u610f\u4e00\u70b9\uff08\u4e0d \u4e0e\u70b9E\u3001\u70b9C\u91cd\u5408\uff09\u65f6\uff0c\u5176\u5b83\u6761\u4ef6\u4e0d\u53d8\uff0c\u5219\uff081\uff09\u4e2d\u7684\u7ed3\u8bba\u662f\u5426\u4ecd\u7136\u6210\u7acb\uff1f\u82e5\u6210\u7acb\uff0c\u8bf7\u7ed9\u4e88\u8bc1\u660e\uff1b\u82e5\u4e0d\u6210\u7acb\uff0c\u8bf7\u8bf4\u660e\u7406\u7531\uff0e
\uff083\uff09\u5982\u56fe3\uff0c\u5f53\u70b9P\u4e3a\u7ebf\u6bb5EC\u5ef6\u957f\u7ebf\u4e0a\u7684\u4efb\u610f\u4e00\u70b9\u65f6\uff0c\u5176\u5b83\u6761\u4ef6\u4e0d\u53d8\uff0c\u5219PR\u4e0ePQ\u4e4b\u95f4\u53c8\u5177\u6709\u600e\u6837\u7684\u6570\u91cf\u5173\u7cfb\uff1f\u8bf7\u76f4\u63a5\u5199\u51fa\u4f60\u7684\u731c\u60f3\uff0e

\u3010\u7b54\u6848\u3011\u89e3\uff1a\uff082\uff09\u56fe2\u4e2d\u7ed3\u8bbaPR\uff0bPQ= \u4ecd\u6210\u7acb\u3002\u8bc1\u660e\u5982\u4e0b\uff1a
\u8fde\u63a5BP\uff0c\u8fc7C\u70b9\u4f5cCK\u22a5BD\u4e8e\u70b9K\u3002
\u2235\u56db\u8fb9\u5f62ABCD\u4e3a\u77e9\u5f62\uff0c\u2234\u2220BCD=90\u00b0\u3002
\u53c8\u2235CD=AB=3\uff0cBC=4\uff0c\u2234 \u3002
\u2235S\u25b3BCD= BC•CD= BD•CK\uff0c\u22343\u00d74=5CK\uff0c\u2234CK= \u3002
\u2235S\u25b3BCE= BE•CK\uff0cS\u25b3BEP= PR•BE\uff0cS\u25b3BCP= PQ•BC\uff0c\u4e14S\u25b3BCE=S\u25b3BEP\uff0bS\u25b3BCP\uff0c
\u2234 BE•CK= PR•BE\uff0b PQ•BC\u3002
\u53c8\u2235BE=BC\uff0c\u2234 CK= PR\uff0b PQ\u3002\u2234CK=PR\uff0bPQ\u3002
\u53c8\u2235CK= \uff0c\u2234PR\uff0bPQ= \u3002
\uff083\uff09\u56fe3\u4e2d\u7684\u7ed3\u8bba\u662fPR\uff0dPQ= \uff0e
\u3010\u8003\u70b9\u3011\u77e9\u5f62\u7684\u6027\u8d28\uff0c\u4e09\u89d2\u5f62\u7684\u9762\u79ef\uff0c\u52fe\u80a1\u5b9a\u7406\u3002
\u3010\u5206\u6790\u3011\uff082\uff09\u8fde\u63a5BP\uff0c\u8fc7C\u70b9\u4f5cCK\u22a5BD\u4e8e\u70b9K\uff0e\u6839\u636e\u77e9\u5f62\u7684\u6027\u8d28\u53ca\u52fe\u80a1\u5b9a\u7406\u6c42\u51faBD\u7684\u957f\uff0c\u6839\u636e\u4e09\u89d2\u5f62\u9762\u79ef\u76f8\u7b49\u53ef\u6c42\u51faCK\u7684\u957f\uff0c\u6700\u540e\u901a\u8fc7\u7b49\u91cf\u4ee3\u6362\u5373\u53ef\u8bc1\u660e\u3002
\uff083\uff09\u56fe3\u4e2d\u7684\u7ed3\u8bba\u662fPR\uff0dPQ=125 \u3002
\u8fde\u63a5BP\uff0cS\u25b3BPE\uff0dS\u25b3BCP=S\u25b3BEC\uff0cS\u25b3BEC \u662f\u56fa\u5b9a\u503c\uff0cBE=BC \u4e3a\u4e24\u4e2a\u5e95\uff0cPR\uff0cPQ \u5206\u522b\u4e3a\u9ad8\uff0c\u4ece\u800cPR\uff0dPQ= \u3002
\uff082012\u5c71\u4e1c\u5fb7\u5dde12\u5206\uff09\u5982\u56fe\u6240\u793a\uff0c\u73b0\u6709\u4e00\u5f20\u8fb9\u957f\u4e3a4\u7684\u6b63\u65b9\u5f62\u7eb8\u7247ABCD\uff0c\u70b9P\u4e3a\u6b63\u65b9\u5f62AD\u8fb9\u4e0a\u7684\u4e00\u70b9\uff08\u4e0d\u4e0e\u70b9A\u3001\u70b9D\u91cd\u5408\uff09\u5c06\u6b63\u65b9\u5f62\u7eb8\u7247\u6298\u53e0\uff0c\u4f7f\u70b9B\u843d\u5728P\u5904\uff0c\u70b9C\u843d\u5728G\u5904\uff0cPG\u4ea4DC\u4e8eH\uff0c\u6298\u75d5\u4e3aEF\uff0c\u8fde\u63a5BP\u3001BH\uff0e
\uff081\uff09\u6c42\u8bc1\uff1a\u2220APB=\u2220BPH\uff1b
\uff082\uff09\u5f53\u70b9P\u5728\u8fb9AD\u4e0a\u79fb\u52a8\u65f6\uff0c\u25b3PDH\u7684\u5468\u957f\u662f\u5426\u53d1\u751f\u53d8\u5316\uff1f\u5e76\u8bc1\u660e\u4f60\u7684\u7ed3\u8bba\uff1b
\uff083\uff09\u8bbeAP\u4e3ax\uff0c\u56db\u8fb9\u5f62EFGP\u7684\u9762\u79ef\u4e3aS\uff0c\u6c42\u51faS\u4e0ex\u7684\u51fd\u6570\u5173\u7cfb\u5f0f\uff0c\u8bd5\u95eeS\u662f\u5426\u5b58\u5728\u6700\u5c0f\u503c\uff1f\u82e5\u5b58\u5728\uff0c\u6c42\u51fa\u8fd9\u4e2a\u6700\u5c0f\u503c\uff1b\u82e5\u4e0d\u5b58\u5728\uff0c\u8bf7\u8bf4\u660e\u7406\u7531\uff0e

\u3010\u7b54\u6848\u3011\u89e3\uff1a\uff081\uff09\u5982\u56fe1\uff0c\u2235PE=BE\uff0c\u2234\u2220EBP=\u2220EPB\uff0e
\u53c8\u2235\u2220EPH=\u2220EBC=90\u00b0\uff0c
\u2234\u2220EPH\ufe63\u2220EPB=\u2220EBC\ufe63\u2220EBP\uff0c\u5373\u2220PBC=\u2220BPH\u3002
\u53c8\u2235AD\u2225BC\uff0c\u2234\u2220APB=\u2220PBC\u3002\u2234\u2220APB=\u2220BPH\u3002
\uff082\uff09\u25b3PHD\u7684\u5468\u957f\u4e0d\u53d8\u4e3a\u5b9a\u503c8\u3002\u8bc1\u660e\u5982\u4e0b\uff1a
\u5982\u56fe2\uff0c\u8fc7B\u4f5cBQ\u22a5PH\uff0c\u5782\u8db3\u4e3aQ\u3002
\u7531\uff081\uff09\u77e5\u2220APB=\u2220BPH\uff0c
\u53c8\u2235\u2220A=\u2220BQP=90\u00b0\uff0cBP=BP\uff0c
\u2234\u25b3ABP\u224c\u25b3QBP\uff08AAS\uff09\u3002\u2234AP=QP\uff0cAB=BQ\u3002
\u53c8\u2235AB=BC\uff0c\u2234BC=BQ\u3002
\u53c8\u2235\u2220C=\u2220BQH=90\u00b0\uff0cBH=BH\uff0c
\u2234\u25b3BCH\u224c\u25b3BQH\uff08HL\uff09\u3002\u2234CH=QH\u3002
\u2234\u25b3PHD\u7684\u5468\u957f\u4e3a\uff1aPD+DH+PH=AP+PD+DH+HC=AD+CD=8\u3002
\uff083\uff09\u5982\u56fe3\uff0c\u8fc7F\u4f5cFM\u22a5AB\uff0c\u5782\u8db3\u4e3aM\uff0c\u5219FM=BC=AB\u3002
\u53c8\u2235EF\u4e3a\u6298\u75d5\uff0c\u2234EF\u22a5BP\u3002
\u2234\u2220EFM+\u2220MEF=\u2220ABP+\u2220BEF=90\u00b0\u3002\u2234\u2220EFM=\u2220ABP\u3002
\u53c8\u2235\u2220A=\u2220EMF=90\u00b0\uff0cAB=ME\uff0c\u2234\u25b3EFM\u224c\u25b3BPA\uff08ASA\uff09\u3002
\u2234EM=AP=x\uff0e
\u2234\u5728Rt\u25b3APE\u4e2d\uff0c\uff084\ufe63BE\uff092+x2=BE2\uff0c\u5373 \u3002
\u2234 \u3002
\u53c8\u2235\u56db\u8fb9\u5f62PEFG\u4e0e\u56db\u8fb9\u5f62BEFC\u5168\u7b49\uff0c
\u2234 \u3002
\u2235 \uff0c\u2234\u5f53x=2\u65f6\uff0cS\u6709\u6700\u5c0f\u503c6\u3002
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\u5f535\uff1cx\u22649\u65f6\uff0c\u5982\u56fe3\uff0c

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(1)\u8fde\u63a5AE\uff0c\u5f53\u25b3APE\u4e0e\u25b3ADE\u5168\u7b49\u65f6\uff0c\u6c42BP\u7684\u957f\uff1b
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(3)\u82e5PE\u2225BD\uff0c\u8bd5\u6c42\u51fa\u6b64\u65f6BP\u7684\u957f.

\u3010\u7b54\u6848\u3011\u89e3\uff1a\uff081\uff09\u2235\u25b3APE\u224c\u25b3ADE\uff0c\u2234AP=AD=3\u3002
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\u2234 \uff0c\u5373\u3002\u2234\u3002
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\uff082\uff09\u8bbeBP=x, \u7531\uff082\uff09\u5f97\u3002
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\u2234\u5f53BP= \u65f6\uff0c PE\u2225BD\u3002

看图。

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