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What is the perimeter of a rectangle?

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Math Revolution GMAT Instructor
Joined: 16 Aug 2015
Posts: 7999
GMAT 1: 760 Q51 V42
GPA: 3.82
What is the perimeter of a rectangle?  [#permalink]

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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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"Only $79 for 1 month Online Course" "Free Resources-30 day online access & Diagnostic Test" "Unlimited Access to over 120 free video lessons - try it yourself" CEO Joined: 12 Sep 2015 Posts: 3990 Location: Canada Re: What is the perimeter of a rectangle? [#permalink] Show Tags 20 Sep 2018, 06:43 Top Contributor 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 RELATED VIDEO _________________ Test confidently with gmatprepnow.com Math Expert Joined: 02 Aug 2009 Posts: 7953 Re: What is the perimeter of a rectangle? [#permalink] Show Tags 20 Sep 2018, 07:02 1 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 _________________ GMATH Teacher Status: GMATH founder Joined: 12 Oct 2010 Posts: 935 Re: What is the perimeter of a rectangle? [#permalink] Show Tags 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. Attachments 20Set18_5m.gif [ 22.84 KiB | Viewed 461 times ] _________________ Fabio Skilnik :: GMATH method creator (Math for the GMAT) Our high-level "quant" preparation starts here: https://gmath.net Math Revolution GMAT Instructor Joined: 16 Aug 2015 Posts: 7999 GMAT 1: 760 Q51 V42 GPA: 3.82 Re: What is the perimeter of a rectangle? [#permalink] Show Tags 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 _________________ MathRevolution: Finish GMAT Quant Section with 10 minutes to spare The one-and-only World’s First Variable Approach for DS and IVY Approach for PS with ease, speed and accuracy. "Only$79 for 1 month Online Course"
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Re: What is the perimeter of a rectangle?   [#permalink] 24 Sep 2018, 05:18
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