Diagonals of the rhombus

How long are the diagonals e, and f in the diamond if its side is 5 cm long and its area is 20 cm2?

Correct answer:

e =  4.4721 cm
f =  8.9443 cm

Step-by-step explanation:

a=5 cm S=20 cm2  S = a h h=S/a=20/5=4 cm  h = a   sin(A) A=π180°arcsin(h/a)=π180°arcsin(4/5)53.1301   e=a 22 cosA=a 22 cos53.130102354156° =5 22 cos53.130102354156° =5 22 0.6=4.472=4.4721 cm
f=a 2+2 cosA=a 2+2 cos53.130102354156° =5 2+2 cos53.130102354156° =5 2+2 0.6=8.94427=8.9443 cm   Verifying Solution:  S = EF/2 (E/2)2 + (F/2)2 = a2  (E/2)2 + (2 S/E/2)2 = a2 E2 + 4 S2/E2 = 4 a2 z = E2  z + 4 S2/z = 4 a2  z2+4 S2=4 a2 z  z2+4 202=4 52 z z2100z+1600=0  a=1;b=100;c=1600 D=b24ac=1002411600=3600 D>0  z1,2=2ab±D=2100±3600 z1,2=2100±60 z1,2=50±30 z1=80 z2=20  z=z1=80  E=z=80=4 5 cm8.9443 cm F=2 S/E=2 20/8.9443=2 5 cm4.4721 cm

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