BdMO National Junior 2010/2
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samiul_samin
- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
A rectangle and a square have the same area,find with proof,which one has a greater perimeter?
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samiul_samin
- Posts:1007
- Joined:Sat Dec 09, 2017 1:32 pm
Re: BdMO National Junior 2010/2
Answer:
\[Rectangle\]
Proof:
Suppose the area is $c^2$
$a, b$ are two sides of a rectangle and $c$ is the side of the square.
Acoording the question:
\[a\times b=c\times c\]
If $2(a+b)=2(c)$ then $(a-b)^2=0$ and the reactangle will turn into a square.
Else if $(a+b)<2c$ then $(a-b)^2<0$ and it is not possible .
So,$(a+b)>2c\Rightarrow 2(a+b)>4c$
$[Q.E.D]$
\[Rectangle\]
Proof:
Suppose the area is $c^2$
$a, b$ are two sides of a rectangle and $c$ is the side of the square.
Acoording the question:
\[a\times b=c\times c\]
If $2(a+b)=2(c)$ then $(a-b)^2=0$ and the reactangle will turn into a square.
Else if $(a+b)<2c$ then $(a-b)^2<0$ and it is not possible .
So,$(a+b)>2c\Rightarrow 2(a+b)>4c$
$[Q.E.D]$