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14 Jun 2021, 23:55
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It is a square iff the four sides are the same.
This means that if it is a square, the perimeter should be 4 * length of one side and that the area should be one of the equal sides squared.
This means the area of a square must be (P/4)^2
A = (P/4)^2
Now, let us take the statements and see if we can make either one of them look like A = (P/4)^2
Statement 1: P = 4/3 A => (3/4)P = A... this doesn't look right
Statement 2: P = 4*sqrt(A) => p/4 = sqrt(A) => A = (p/4)^2... this works!
Therefore, only statement one is will tell you the right answer.
Someone confirm my method?
This means that if it is a square, the perimeter should be 4 * length of one side and that the area should be one of the equal sides squared.
This means the area of a square must be (P/4)^2
A = (P/4)^2
Now, let us take the statements and see if we can make either one of them look like A = (P/4)^2
Statement 1: P = 4/3 A => (3/4)P = A... this doesn't look right
Statement 2: P = 4*sqrt(A) => p/4 = sqrt(A) => A = (p/4)^2... this works!
Therefore, only statement one is will tell you the right answer.
Someone confirm my method?
15 Jun 2021, 00:25
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hassan233
It is a square iff the four sides are the same.
This means that if it is a square, the perimeter should be 4 * length of one side and that the area should be one of the equal sides squared.
This means the area of a square must be (P/4)^2
A = (P/4)^2
Now, let us take the statements and see if we can make either one of them look like A = (P/4)^2
Statement 1: P = 4/3 A => (3/4)P = A... this doesn't look right
Statement 2: P = 4*sqrt(A) => p/4 = sqrt(A) => A = (p/4)^2... this works!
Therefore, only statement one is will tell you the right answer.
Someone confirm my method?
This means that if it is a square, the perimeter should be 4 * length of one side and that the area should be one of the equal sides squared.
This means the area of a square must be (P/4)^2
A = (P/4)^2
Now, let us take the statements and see if we can make either one of them look like A = (P/4)^2
Statement 1: P = 4/3 A => (3/4)P = A... this doesn't look right
Statement 2: P = 4*sqrt(A) => p/4 = sqrt(A) => A = (p/4)^2... this works!
Therefore, only statement one is will tell you the right answer.
Someone confirm my method?
upon further investigation of my own work, this will NOT work!
What I wrote for statement 1 shows that it is possible for it NOT to be a square, but does not show whether it CAN be a square. I need to do a 2-way proof (that is, show that with that equation it CAN be a square and can also NOT be a square). I did half of that. Not good enough because if it is NOT a square, it would still be sufficient since I would be able to answer the question. I need to prove that it does NOT have the potential to become a square, not just proof that it is possible that it is not a square.
For statement one, the work I showed still makes it clear that it can only be a square, since the four parameters would HAVE to be the same for the equation to work.
16 Jun 2021, 19:28
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Reinfrank2011
olifurlong
If a rectangular region has perimeter \(P\) inches and area \(A\) square inches, is the region square?
(1) \(P = \frac{4}{3}*A\)
(2) \(P = 4\sqrt{A}\)
(1) \(P = \frac{4}{3}*A\)
(2) \(P = 4\sqrt{A}\)
Let the rectangular region have sides \(x\) and \(y\).
--> \(Perimeter = P = 2(x+y)\)
--> \(Area = A = xy\)
Question: Is \(x = y\) ?
(1) \(P = \frac{4}{3}*A\)
\(2(x+y) = (\frac{4}{3})xy\)
\(\frac{x+y}{xy} = \frac{4}{6} = \frac{2}{3}\)
\(\frac{1}{x} + \frac{1}{y} = \frac{4}{6} = \frac{2}{3}\)
So we have that the sum of two values equals \(\frac{2}{3}\)
If \(\frac{1}{x} = \frac{1}{y} = \frac{1}{3}\), then the answer is YES
If \(\frac{1}{x} = \frac{1}{6}\) and \(\frac{1}{y} =\frac{3}{6}\), then the answer is NO
Two different answers --> INSUFFICIENT
(2) \(P = 4\sqrt{A}\)
\(2(x + y) = 4\sqrt{xy}\)
\(x + y = 2\sqrt{xy}\)
\(x + y - 2\sqrt{xy} = 0\)
\((\sqrt{x} - \sqrt{y})^2 = 0\)
\(\sqrt{x} - \sqrt{y} = 0\)
\(\sqrt{x} = \sqrt{y}\)
\(x = y\)
--> SUFFICIENT
In statement 2; if all we have proved is x=y , then it could be a rhombus too right? Without knowing if the angles are 90 degrees, how can we be certain?
