Răspuns:
Explicație pas cu pas:
a. [tex]A_{AMNC}=A_{ABCD}-A_{ABM}-A_{ADN}[/tex]
[tex]A_{ABCD}=l^2[/tex]
DC=l
[tex]DN=DC-NC=DC-\frac{DC}{3} =\frac{2l}{3} =BM[/tex]
[tex]A_{ADN}=A_{ABM}=\frac{l\cdot \frac{2l}{3} }{2} =\frac{l^2}{3}[/tex]
[tex]A_{AMNC}=l^2-\frac{l^2}{3} -\frac{l^2}{3} =\frac{l^2}{3}[/tex]
b. Fie MP⊥AN
Facem prin aria in 2 moduri in ΔAMN
[tex]A_{MNC}=\frac{MC\cdot CN}{2} =\frac{l^2}{18}[/tex]
[tex]A_{AMN}=A_{AMNC}-A_{MNC}=\frac{l^2}{3} -\frac{l^2}{18} =\frac{5l^2}{18} =\frac{MP\cdot AN}{2}[/tex]
Aflam AN inΔAND aplicand Pitagora
AN²=AD²+DN²
[tex]AN^2=l^2+\frac{4l^2}{9} =\frac{13l^2}{9} \\\\AN=\frac{l\sqrt{13} }{3}[/tex]
[tex]A_{AMN}=\frac{5l^2}{18} =\frac{MP\cdot AN}{2}\\\\\frac{5l^2}{18} =\frac{MP\cdot l\sqrt{13} }{6} \\\\MP=\frac{5l}{3\sqrt{13} } =\frac{5l\sqrt{13} }{39}[/tex]
Sa mai verifici odata tu calculele
