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L=length; W=width; A=area; D=diagonal
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LW=A
LW=50
L=50/W
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L^2+W^2=D^2
(50/W)^2+W^2=(sqrt(125))^2
2500/W^2+W^2=125 Multiply each side by W^2
2500+W^4=125W^2
W^4-125W^2+2500=0
Let X=W^2:
X^2-125X+2500=0
(X-25)(X-100)=0
(X-25=0) OR (X-100=0)
{X=25 OR X=100 Substitute W^2 for X.
W^2=25 OR W^2=100
W=5 or W=10
ANSWER 1: The width is 5 or 10 units.
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If W=5 units:
L=50/W=50/5=10 units
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If W=10 units:
L=50/W=50/10=5 units
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ANSWER (DIMENSIONS):
Length is 10 units and width is 5 units,
or width is 10 units and length is 5 units.
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Perimeter is same in both cases:
Perimeter=2(L+W)=2(10+5)=2(15)=30 units
ANSWER (PERIMETER): The perimeter is 30 units.
