Given is a rectangle with perimeter $x$ cm and side lengths in a $1:2$ ratio. Suppose that the area of the rectangle is also $x$ $\text{cm}^2$. Determine all possible values of $x$.
Problem
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Tags: geometry
joshualiu315
21.05.2023 21:43
Let the sidelengths be $s$ and $2s$.
We have $x=6s=2s^2$, so $s^2-3s=0$, so $s=0,3$ but assuming the rectangle is non-degenerate, we have $s=3$, so $x=18$.
JanHaj
27.06.2023 22:52
Let $a,b$ be the side lengths of this rectangle, such that $a=2b$. Now,knowing that the perimeter of a rectangle is $2(a+b)$, its area is $ab$ and that they are equal to $x$ (eachother) we have: $x=2(a+b)=ab$ $<=>$ $2\cdot 3b=2b\cdot b$ $<=>$ $6b=2b^2$ $<=>$ $2b=6$ $<=>$ $b=3$ So $a=2\cdot 3=6$ and $x=ab=6\cdot3=18$
Matematik1106
27.10.2024 16:14
P=2(a+b)=x S=ab=x 2(a+b)=ab a=2b 2×3b=2b×b b=3 a=6 x= a×b=3×6=18 x=18
lksb
28.10.2024 05:35
$$A=2l^2=x$$$$P=6l=x$$$$2l^2=6l\implies l=3\implies x=6\cdot3=18$$