Question

Triunghiul ABC are latura BC inclusa intr-un plan a si AB =24cm , AC=32 cm si BC=40 cm .Fie E € (AB) si F €(AC) astfel incat AE=8cm si AF=6cm .
a) Daca dreapta EF intersecteaza planul a in P ,aratati ca punctele P, B si C sunt coliniare.
b) Calculati perimetrul patrulaterului BEFC .

Answer 1

BC²=AB²+AC²(se verifica)
atunci 
EF²=AE²+AF²
EF=10 cm
2.ΔABC≈ΔAEF
AE/AC=AF/AB=EF/BC
8/32=6/24=10/40=1/4
atunci
∧AEF≡∧ACB
∧AFE≡∧ABC
Trebuie demonstrat ca PBA+ABC=180°
∧ABC=90°-∧C
PBA=180°-P-∧PEB
∧PEB=∧AEF=∧C
∧PBA=180°-(∧P+∧C)
dar ∧ABC=∧AEF⇒
∧AFE+∧PFC=180°⇒
∧PBA+∧ABC=180°
2.perimetrul
EA=24-8=16
AF=32-6=26
EF=10
P=10+40+16+26=92 cm