Question

1.In trapezul isoscel ABCD cu AB||CD,AB>CD,se cunosc CD=6cm,BC=5cm  si (,ADC)=120°.Se construieste CE_|_ AB,E€(AB).Calculati AE si perimetrul trapezului.
2.In trapezul dreptunghic ABCD,AB||CD,se stie m(<A)=m(<D)=90,m(<B)=45,AC_|_BC,iar CD=12.Aratati ca AB=2CD si calculati aria trapezului.

Answer 1

1.Daca trapezul este isoscel, atunci <ADC=<DCB=120  si <DAB=<ABC=60
=> in ΔEBC  avem cos <B=cos 60 = 1/2=EB/CB=EB/5=> EB=5/2=2.5
AB=2*EB+DC=5+6=11
AE=2.5+6=8.5
Perimetru=6+5+5+11=27

2. in ΔABC, DACA <B=45°=> <BAC=45° => ΔABC=isoscel cu AC=CB
Daca <BAD=90 => <CAD=45°
in ΔADC=> <ACD=45°=> ΔADC este isoscel cu AD=DC
=> AC=CD√2
revenim la ΔABC
AB=AC√2=CD√2*√2= 2CD

Arie trapez= (CD+AB)*AD:2
AC=12√2
AB=2CD=24
AD=CD=12
=> Arie trapez= (12+24)*12:2=36*6=216