Question

1)In triunghiul ABC , cu AB=24 cm , consideram mediana [CD], CD=36 cm. Daca M , N , P apartine [BC] astfel incat BM=MN=PC si M', N', P' apartine [CD] astfel incat MM'||NN'||PP'||AB , calculati lunginile NN'||, PP', DN', M'C.
2)In triunghiul ABC, [AM] este mediana M apartine [BC].Perpendicular in A pe AM intersecteaza paralele prin B si C la AM in punctele N si P. Demonstrati ca [AN] congruent cu [AP]

Answer 1

1)
BM≡MN≡NP≡PC⇒BM+MN=NP+PC⇒BN≡NC
NN'||AB⇒NN' l.m in ΔADC
BD=AB/2 (CD mediana)=24/2=12
NP≡PC⇒NN' l,m in Δ ADC, NN'=12/2=6

PP' l.m inΔ ANN', PP'=NN'/2=6/2=3

DN'=(Thales) (BN/BC)*DC=(1/2)DC=36/2=18*

CM'=(Thales)=(CM/CB)*CD=(3/4)*CD=(3/4) *36=27


2)
Fie d⊥AM, A∈d
BN||AM||CP⇒BNCPtrapez (dreptunghic) si AM linie mijlocie (BM≡MC)⇒AN≡CP
altfel
BN||AM||CP⇒(fascicol de drepte paralele) AN/NP=BM/MC=1⇒AN=NP