Step-by-step explanation:
say, we call
x = orange length
y = orange width
the area of the blue rectangle is
9×20 = 180 m²
as they have the same area :
orange length × orange width = 180 m²
and then we know the perimeter of the orange rectangle :
2×orange length + 2×orange width = 54 m
simplified :
orange length + orange width = 27 m
so, we know already just by experience and by looking at the numbers :
15 m × 12 m
but how to elaborate it formally ?
by solving the 2 equations in 2 variables :
orange length × orange width = 180
orange length + orange width = 27
orange length = 27 - orange width
and we use that in the first equation :
(27 - orange width) × orange width = 180
27×orange width - orange width² = 180
-orange width² + 27×orange width - 180 = 0
the general solution to a quadratic equation
ax² + bx + c = 0
is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
orange width = x
a = -1
b = 27
c = -180
x = (-27 ± sqrt(27² - 4×-1×-180))/(2×-1) =
= (-27 ± sqrt(729 - 720))/-2 =
= (-27 ± sqrt(9))/-2 = (-27 ± 3)/-2
x1 = (-27 + 3)/-2 = -24/-2 = 12
x2 = (-27 - 3)/-2 = -30/-2 = 15
and here we have it :
orange width = 12 m
and then
orange length = 27 - 12 = 15 m
or
orange width = 15 m
and then
orange length = 27 - 15 = 12 m
both are valid solutions.
simply because of the terms length and width I would pick for length the longer solution (and we usually say length by width).
so, the orange rectangle is
15 meters by 12 meters.