5.
[tex]A=\frac{MP\times NQ}{2} \\\\8\sqrt{6}\times 2= 4\sqrt{6} \times NQ\\\\NQ=4\ cm[/tex]
Fie {O}=MP∩NQ
MO=2√6
NO=2
Aplicam Pitagora in ΔMON dr in O
MN²=24+4=28
MN=2√7 cm
[tex]P_{MNPQ}=4MN=8\sqrt{7} \ cm[/tex]
6.
L=15 cm
l=10 cm
Aplicam Pitagora in ΔBD dr in A
BD²=AD²+AB²
BD²=225+100=325
BD=5√13 cm
P=AB+BD+AD=25+5√13
5√13≅18
25+18=43<45
Fie EF⊥CD, EF║BC
AE bis ⇒ Teorema bisectoarei
[tex]\frac{AD}{AB} =\frac{DE}{EB} \\\\\frac{DE}{EB}=\frac{10}{15}=\frac{2}{3}[/tex]
[tex]\frac{DE}{DE+EB}= \frac{2}{2+3} \\\\\frac{DE}{DB} =\frac{2}{5}[/tex]
EF║BC⇒
[tex]\frac{DE}{DB} =\frac{DF}{DC} =\frac{EF}{BC} \\\\\frac{DE}{DB} =\frac{EF}{BC}\\\\\frac{2}{5} =\frac{EF}{10}[/tex]
EF=4 cm
[tex]\frac{2}{5} =\frac{DF}{15}[/tex]
DF=6 cm⇒FC=9 cm
Aplicam Pitagora in EFC dr in F
EC²=EF²+FC²
EC²=16+81
EF=√97cm
