Question

in trapezul dreptunghic ABCD , m (A) = m(D)=90, AB<CD, iar AB si CD sunt direct proportionale cu numerele 2 si 3. Se stie ca BD perpendicular pe BC iar AD= 3√2 cm
 Determinati AB SI CD
 Calculati aria Trapezului ABCD
 Determinati Lungimile diagonalelor trapezului [AB] si [BD]

Answer 1

ABCD-trapez dreptunghic
mas<A=mas<D=90

BD _|_ BC  <=>mas<DBC=90
ΔDBC
 fie [BN _|_ Dc
BN-h

BN=AD=3√2

AB;DC   dp cu  2;3

AB/2=DC/3=k
AB=2k
DC=3k

DN=AB=2k

NC=DC-DN=3k-2k=k

ΔDBC, mas<B=90
DB²=DN*DC

DB²=2k*3k= 6k² 

ΔBND, mas<N=90
DB²=DN²+BN²

(k√6)²= (2k)² + (3√2)²

6k² =4k² + 18

6k²-4k² =18

2k²=18

k²=9   =>k=3

AB=2*3=6
DC=3*3=9

A ABCD= (AB+DC)*BN/2=  (6+9)*3√2/2=  15 *3√2/2= 7,5 *3√2= 22,5√2

DB²=6k²  =>DB=k√6=  3√6

ΔADC, mas<D=90
AC²=AD²+DC²=  (3√2)² +9² =18+81 = 99  =>AC=3√11