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What is the perimeter of a rectangle?

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What is the perimeter of a rectangle?  [#permalink]

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New post 20 Sep 2018, 01:59
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[Math Revolution GMAT math practice question]

What is the perimeter of a rectangle?

1) The square of the diagonal is \(52\).
2) The area of the rectangle is \(24\).

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Re: What is the perimeter of a rectangle?  [#permalink]

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New post 20 Sep 2018, 06:43
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MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the perimeter of a rectangle?

1) The square of the diagonal is \(52\).
2) The area of the rectangle is \(24\).


Target question: What is the perimeter of a rectangle?
This is a good candidate for rephrasing the target question.
Let x = the length of the rectangle's base
Let y = the length of the rectangle's height
So, the perimeter of the rectangle = 2x + 2y
REPHRASED target question: What is the value of 2x + 2y?

The video below has tips on rephrasing the target question

Statement 1: The square of the diagonal is 52
In other words, (length of diagonal)² = 52
We can create a RIGHT triangle with the base, height and diagonal.
As such, we can apply the Pythagorean Theorem to write: x² + y² = 52
Is this information (x² + y² = 52) enough to determine the value of 2x + 2y?
NO.
There are several values of x and y that satisfy statement 1. Here are two:
Case a: x = 4 and y = 6. Notice that x² + y² = 4² + 6² = 52. In this case, the answer to the REPHRASED target question is 2x + 2y = 8 + 12 = 20
Case b: x = √51 and y = 1. Notice that x² + y² = (√51)² + 1² = 52. In this case, the answer to the REPHRASED target question is 2x + 2y = 2√51 + 2
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: The area of the rectangle is 24
In other words, xy = 24
Is this information enough to determine the value of 2x + 2y?
NO.
There are several values of x and y that satisfy statement 2. Here are two:
Case a: x = 4 and y = 6. Notice that xy = (4)(6) = 24. In this case, the answer to the REPHRASED target question is 2x + 2y = 8 + 12 = 20
Case b: x = 2 and y = 12. Notice that xy = (2)(12) = 24. In this case, the answer to the REPHRASED target question is 2x + 2y = 4 + 24 = 28
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
Statement 1 tells us that x² + y² = 52
Statement 2 tells us that xy = 24, which also means 2xy = 48

Add the two red equations to get: x² + 2xy + y² = 100
Factor the left side to get: (x + y)² = 100
This means that: x + y = 10
Multiply both sides by 2 to get: 2x + 2y = 20
Perfect!!
Since we can answer the REPHRASED target question with certainty, the combined statements are SUFFICIENT

Answer: C

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Re: What is the perimeter of a rectangle?  [#permalink]

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New post 20 Sep 2018, 07:02
What is the perimeter of a rectangle?

we require to know 2(L+B), so we have to know dimensions or SUM of dimensions

1) The square of the diagonal is \(52\).
the square of diagonal is SUM of the square of the dimensions, so \(L^2+B^2=D^2........L^2+B^2=52\) But remember that the dimensions need not be integers
there can be various combinations of L and B satisfying this..
a) L=7 and B=\(\sqrt{3}\).....\(7^2+\sqrt{3}^2=49+3=52\) but \(2(L+B)=2(7+\sqrt{3})\)
b) L=6 and B=4........\(6^2+4^2=36+16=52\) but \(2(L+B)=2(6+4)=20\)
Insuff

2) The area of the rectangle is \(24\).
area = LB = 24
L=6 and B=4 OR L=12 and B = 2
different cases
Insuff

combined..
\(L^2+B^2=52...........(L+B)^2-2LB=52\) but statement II tells us that LB is 24
so \((L+B)^2-48=52........(L+B)^2=100........\)
but perimeter can only be positive, so L+B=10 and perimeter = 2*10=20
sufficient

C
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Re: What is the perimeter of a rectangle?  [#permalink]

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New post 20 Sep 2018, 09:52
MathRevolution wrote:
[Math Revolution GMAT math practice question]

What is the perimeter of a rectangle?

1) The square of the diagonal is \(52\).
2) The area of the rectangle is \(24\).


\(? = {\text{perim}}\left( {{\text{rectangle}}} \right)\)

Excellent opportunity to GEOMETRICALLY BIFURCATE each statement alone:

\(\left( 1 \right)\,\,\,{\text{dia}}{{\text{g}}^{\,{\text{2}}}} = 52\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{diag}}\,\, > \,\,0} \,\,\,{\text{diag}}\,\,{\text{unique}}\,\,\,{\text{but}}\,\,\,{\text{INSUFF}}.\)

\(\left( 2 \right)\,\,\,{\text{area}} = 24\,\,\,\,\, \Rightarrow \,\,\,\,{\text{INSUFF}}{\text{.}}\)

(See the image attached!)

\(\left( {1 + 2} \right)\)

Let L and W be the length and width of our focused-rectangle. Hence:

\(? = {\text{2}}\left( {L + W} \right)\)

\(\left( {1 + 2} \right)\,\,\left\{ \begin{gathered}
{L^2} + {W^2} = 52 \hfill \\
2LW = 2 \cdot 24\,\,\,\, \hfill \\
\end{gathered} \right.\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,{\left( {L + W} \right)^2} = 52 + 48 = 100\)

\({\left( {L + W} \right)^2} = 100\,\,\,\,\mathop \Rightarrow \limits^{L + W\,\, > \,\,0} \,\,\,\,L + W\,\,\,{\text{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = 2\left( {L + W} \right)\,\,\,\,{\text{unique}}\)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.
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Re: What is the perimeter of a rectangle?  [#permalink]

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New post 24 Sep 2018, 05:18
=>

Forget conventional ways of solving math qAnswer: CAnswer: Cuestions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

When we apply VA method to geometry, we need to count the number of variables. For a rectangle, we have two variables for the length and the width of the rectangle. Let x and y be the length of the width of the rectangle, respectively.

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2)
We have \(x^2+y^2 = 52\) by Pythagoras’ theorem, and \(Area = xy = 24.\)
So, \((x+y)^2 = x^2+2xy + y^2 = (x^2+y^2) +2xy = 52 + 48 = 100.\)
Therefore, \(x+y = 10\) and we can calculate the perimeter.
Both conditions (together) are sufficient.

Therefore, C is the answer.
Answer: C
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Re: What is the perimeter of a rectangle? &nbs [#permalink] 24 Sep 2018, 05:18
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