Salut! =)
ABCD - trapez isoscel ;
AB = 4 cm ;
CD = 8 cm ;
AD = BC = 4 cm ;
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m ( ∡ C ) = ?
d = ?
AO = ?
a )
Construim BM ⊥ DC
ΔBMC
cos ( ∡ C ) = Ca / Ip
cos ( ∡ C ) = CM / CB
cos ( ∡ C ) = 2 / 4
cos ( ∡ C ) = 1 / 2
=> m ( ∡ C ) = 60°
b )
Δ BMC - dreptunghic ;
BM² = BC² - MC²
BM² = 16 - 4
BM² = 12
BM = 2√3 cm
ΔBMD - dreptunghic ;
BD² = BM² + DM²
BD² = 12 + 36
BD = √48
BD = 4√3 cm ;
c )
AN ⊥ BC
m ( ∡ NAB ) = 90°
m ( ∡ NAC ) = 60° } = > m ( ∡NAC ) = 30°
m ( ∡ MBC ) = 30°
m ( ∡ MBD ) = 60°
m ( ∡ MBD ) + m ( ∡ DBA ) = 90°
m ( ∡ DBA ) = 30°
=> ΔAOB - isoscel ( ∡OAB = ∡OBA )
m ( ∡ AOB ) = 120°
m ( ∡ AOB ) + m ( ∡ BOC ) = 180°
m ( ∡ BOC ) = 180° - 120°
m ( ∡ BOC ) = 60°
m ( ∡ MBD ) = 60° } = > ΔBOP - echilateral
P = AC ∩ BM
AO = OB
OB = OP
=> AO = OP = PC
=> AO = AC / 3
AO = 4√3 / 3
Bafta! =)
#COPACEIBRAINLY
