如图,已知:△ABC中,AB=5,BC=4,AC=3PQ//AB,P点在AC上(与点A、C不重合),Q点在BC上.(l)当△PQ当△PQC的面积与四边形PABQ的面积相等时,求CP的长;(2)当△PQC的周长与四边形PABQ的周长相等时,
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![如图,已知:△ABC中,AB=5,BC=4,AC=3PQ//AB,P点在AC上(与点A、C不重合),Q点在BC上.(l)当△PQ当△PQC的面积与四边形PABQ的面积相等时,求CP的长;(2)当△PQC的周长与四边形PABQ的周长相等时,](/uploads/image/z/5340075-51-5.jpg?t=%E5%A6%82%E5%9B%BE%2C%E5%B7%B2%E7%9F%A5%EF%BC%9A%E2%96%B3ABC%E4%B8%AD%2CAB%EF%BC%9D5%2CBC%EF%BC%9D4%2CAC%EF%BC%9D3PQ%2F%2FAB%2CP%E7%82%B9%E5%9C%A8AC%E4%B8%8A%EF%BC%88%E4%B8%8E%E7%82%B9A%E3%80%81C%E4%B8%8D%E9%87%8D%E5%90%88%EF%BC%89%2CQ%E7%82%B9%E5%9C%A8BC%E4%B8%8A%EF%BC%8E%EF%BC%88l%EF%BC%89%E5%BD%93%E2%96%B3PQ%E5%BD%93%E2%96%B3PQC%E7%9A%84%E9%9D%A2%E7%A7%AF%E4%B8%8E%E5%9B%9B%E8%BE%B9%E5%BD%A2PABQ%E7%9A%84%E9%9D%A2%E7%A7%AF%E7%9B%B8%E7%AD%89%E6%97%B6%2C%E6%B1%82CP%E7%9A%84%E9%95%BF%EF%BC%9B%EF%BC%882%EF%BC%89%E5%BD%93%E2%96%B3PQC%E7%9A%84%E5%91%A8%E9%95%BF%E4%B8%8E%E5%9B%9B%E8%BE%B9%E5%BD%A2PABQ%E7%9A%84%E5%91%A8%E9%95%BF%E7%9B%B8%E7%AD%89%E6%97%B6%2C)
如图,已知:△ABC中,AB=5,BC=4,AC=3PQ//AB,P点在AC上(与点A、C不重合),Q点在BC上.(l)当△PQ当△PQC的面积与四边形PABQ的面积相等时,求CP的长;(2)当△PQC的周长与四边形PABQ的周长相等时,
如图,已知:△ABC中,AB=5,BC=4,AC=3PQ//AB,P点在AC上(与点A、C不重合),Q点在BC上.(l)当△PQ
当△PQC的面积与四边形PABQ
的面积相等时,求CP的长;(2)当△PQC的周长与四边形PABQ的周长相等时,求CP的长;(3)试问:在AB上是否存在点M,使得△PQM为等腰直角三角形?若不存在,请简要说明理由;若存在,请求出PQ的长.
如图,已知:△ABC中,AB=5,BC=4,AC=3PQ//AB,P点在AC上(与点A、C不重合),Q点在BC上.(l)当△PQ当△PQC的面积与四边形PABQ的面积相等时,求CP的长;(2)当△PQC的周长与四边形PABQ的周长相等时,
(1)因为△ABC中的AB=5,BC=4,AC=3,
∴△ABC是直角三角形,面积S=3×4÷2=6.
当△PQC面积=四边形PABQ面积时,
△PQC面积S=3.
设PC=3x,QC=4x,
S=3x×4x÷2=3
x²=1/2,
x=√2/2.
CP=3x=3√2/2.
(2)设PC=3x,PA=3-3x,
QC=4x,QB=4-4x,
由PQ是公共边,∴3x+4x=3-3x+4-4x+5
x=6/7,∴CP=3x=16/7.
(3)①当PM=PQ,且PM⊥PQ时:
由PC=3x,QC=4x,∴PQ=PM=5x,
∵△PCQ∽△AMP,
∴AP/PQ=PM/QC,
(3-3x)/5x=5x/4x.
x=12/37,PQ=5x=60/37.
②当QM=PQ,QM⊥PQ时:
BQ/PQ=QM/PC,
(4-4x)/5x=5x/3x
x=12/37,PQ=5x=60/37.
③当PM=QM,且PM⊥QM时:
M到PQ距离MN是PQ的一半,(将MN平移到MP(N到P)
(3-3x)/5x=2.5x/4x,
x=24/29,PQ=5x=120/29.