a)
ΔNAB isoscel, NA=NB
A=8√2 cm²
Fie NN'⊥AB
Notam AB=a
[tex]A_{NAB}=\frac{NN'\times AB}{2}\\\\ 8\sqrt{2}=\frac{NN'\times a}{2} \\\\NN'=\frac{16\sqrt{2} }{a}[/tex]
Fie NM⊥DC
NM=N'M=a
In ΔNN'M, avem Pitagora
NN'²=MN'²+NM²
NN'²=a²+a²
[tex]\frac{512}{a^2} =2a^2\\\\512=2a^4\\\\256=a^4\\\\a=4\ cm[/tex]
a= 4 cm
[tex]A_t=6a^2=6\times 16=96\ cm^2\\\\V=a^3=64\ cm^3[/tex]
b)
AC=a√2=4√2 cm
NC²=NC'²+CC'²
NC²=4+16=20
NC=2√5 cm
[tex]NN'=\frac{16\sqrt{2} }{a} =4\sqrt{2} \ cm[/tex]
In ΔANN' dr in N'
AN²=NN'²+AN'²
AN²=32+4=36
AN=6 cm
NM⊥DC
DC⊂(ABCD)
MO⊥AC
AC⊂(ABCD)⇒ NO⊥AC
MO=2 cm
NO²=NM²+OM²
NO²=16+4=20
NO=2√5 cm
[tex]A_{ANC}=\frac{NO\times AC}{2} =\frac{2\sqrt{5} \times 4\sqrt{2} }{2} =4\sqrt{10} \ cm^2[/tex]
