Question

Fie triunghiul ABC cu m(∡A)=90° si AD⊥BC, D∈BC.
a) Daca AB=9cm si BD=3cm, calculati BC.
b) Daca AC=8cm si BC=16cm, calculati CD.
c) Daca BD=21cm si BC=84cm, calculati AB si AC.
d) Daca BD= o patrimedin BC si AB 12 cm, calculati BC si AC.
e) Daca DC/BC=1/4 si AC=6cm, calculati BC si AB.
f) Daca BD/DC=2/7 si BC=18cm, calculati BD, DC, AB, AC.
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Answer 1

folosim teorema catetei
a)AB=9cm si BD=3cm, calculati BC
AB²=BD.BC
81=3.BC
BC=81/3=27 cm
b) Daca AC=8cm si BC=16cm, calculati CD
AC²=DC.BC
64=16.DC
CD=64:16=4 cm
c) Daca BD=21cm si BC=84cm, calculati AB si AC
AB²=BD.BC
AB²=21.84=1764
AB=42 cm
AC²=DC.CB
DC=84-21=63 cm
AC²=63.84=5292
AC=42√3cm
d) Daca BD= o patrime din BC si AB =12 cm, calculati BC si AC.
AB²=BD.BC
144=BC/4.BC
BC²=144.4
BC=12.2=24 cm
BD=24/4=6cm
DC=24-6=18 cm
AC²=DC.BC
AC²=18.24=432
AC=12√3 cm
e) Daca DC/BC=1/4 si AC=6cm, calculati BC si AB.
BC=4.DC
AC²=BC.DC
AC²=4DC.DC=4DC²
AC=2DC
2DC=6⇒DC=3 cm
BC=4.3=12 cm
BD=12-3=9 cm
AB²=BD.BC
AB²=9.12=108
AB=6√3 cm
f) Daca BD/DC=2/7 si BC=18cm, calculati BD, DC, AB, AC.
BD=2DC/7
BC=BD+DC
18=2DC/7+DC=9DC/7
DC=18.7/9= 14 cm
BD=BC-DC=18-14=4 cm
AB²=BD.BC=4.18=72
AB=6√2 cm
AC²=DC.BC
AC²=14.18=252
AC=6√7 cm