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100 points for $49 worth of Veritas practice GMATs FREE VERITAS PRACTICE GMAT EXAMS Earn 10 Points Per Post Earn 10 Points Per Thanks Earn 10 Points Per Upvote ## In the diagram, what is the value of x? tagged by: AAPL ##### This topic has 2 expert replies and 0 member replies ### Top Member ## In the diagram, what is the value of x?  Zoom here, https://postimg.cc/image/u12yw856d/ In the diagram, what is the value of x? $$A.\ 1+\sqrt{2}$$ $$B.\ 1+\sqrt{3}$$ $$C.\ 2\sqrt{2}$$ $$D.\ \sqrt{2}+\sqrt{3}$$ $$E.\ 2\sqrt{3}$$ The OA is B. I don't have clear this PS question. I appreciate if any expert explain it for me. Thank you so much. ### GMAT/MBA Expert GMAT Instructor Joined 04 Oct 2017 Posted: 551 messages Followed by: 11 members Upvotes: 180 Quote: In the diagram, what is the value of x? $$A.\ 1+\sqrt{2}$$ $$B.\ 1+\sqrt{3}$$ $$C.\ 2\sqrt{2}$$ $$D.\ \sqrt{2}+\sqrt{3}$$ $$E.\ 2\sqrt{3}$$ The OA is B. I don't have clear this PS question. I appreciate if any expert explain it for me. Thank you so much. Hi AAPL, Let's take a look at your question. We will solve this problem using Cosine law. Cosine law is $$a^2=b^2+c^2-2bc\ Cos\ A$$ Where a, b and c are the side lengths of a triangle and A is the angle between the sides b and c. Apply Cosine law on the triangle give, we get: $$\left(\sqrt{2}\right)^2=x^2+x^2-2\left(x\right)\left(x\right)\ Cos\ 30$$ $$2=2x^2-2x^2\ Cos\ 30$$ $$1=x^2-x^2\ \frac{\left(\sqrt{3}\right)}{2}$$ $$1=x^2\left(1-\frac{\sqrt{3}}{2}\right)$$ $$1=x^2\left(\frac{2-\sqrt{3}}{2}\right)$$ $$x^2=\frac{2}{2-\sqrt{3}}$$ Let's rationalize the denominator to simplify it. $$x^2=\frac{2}{2-\sqrt{3}}\times\frac{2+\sqrt{3}}{2+\sqrt{3}}$$ $$x^2=\frac{2\left(2+\sqrt{3}\right)}{\left(2\right)^2-\left(\sqrt{3}\right)^2}$$ $$x^2=\frac{2\left(2+\sqrt{3}\right)}{4-3}$$ $$x^2=\frac{2\left(2+\sqrt{3}\right)}{1}$$ $$x^2=4+2\sqrt{3}$$ We need to see if we can write the right hand side as a square of a number? Let's try writing it as a square. $$x^2=1+3+2\sqrt{3}$$ $$x^2=\left(1\right)^2+\left(\sqrt{3}\right)^2+2\left(1\right)\sqrt{3}$$ Using formula a^2+b^2+2ab=(a+b)^2, we can write the above equation as: $$x^2=\left(1+\sqrt{3}\right)^2$$ $$x=\left(1+\sqrt{3}\right)$$ Therefore, Option B is correct. Hope it helps. I am available if you'd like any follow up. _________________ GMAT Prep From The Economist We offer 70+ point score improvement money back guarantee. Our average student improves 98 points. Free 7-Day Test Prep with Economist GMAT Tutor - Receive free access to the top-rated GMAT prep course including a 1-on-1 strategy session, 2 full-length tests, and 5 ask-a-tutor messages. Get started now. ### GMAT/MBA Expert GMAT Instructor Joined 25 May 2010 Posted: 15387 messages Followed by: 1872 members Upvotes: 13060 GMAT Score: 790 AAPL wrote: Zoom here, https://postimg.cc/image/u12yw856d/ In the diagram, what is the value of x? $$A.\ 1+\sqrt{2}$$ $$B.\ 1+\sqrt{3}$$ $$C.\ 2\sqrt{2}$$ $$D.\ \sqrt{2}+\sqrt{3}$$ $$E.\ 2\sqrt{3}$$ The GMAT does not expect knowledge of cosines. This problem can be solved using two common triangles: 45-45-90 and 30-60-90. Step 1: Divide the 30-75-75 triangle into common triangles Step 2: Calculate the sides of 45-45-90 triangle ADC In a 45-45-90 triangle, the sides are in the following ratio: s : s : s√2. Thus, sides of triangle ADC are as follows: AD = 1, DC=1, AC = √2. Step 3: Calculate the sides of 30-60-90 triangle DEC In a 30-60-90 triangle, the sides are in the following ratio: s : s√3 : 2s. In triangle DEC, DC = s√3. Since DC = 1, we get: s√3 = 1 s = 1/√3. Thus: DE = s = 1/√3. CE = 2s = 2/√3. Step 5: Calculate the sides of isosceles triangle ABE The figure for Step 4 implies the following BE = AE = 1 + 1/√3. 6. Determine the value of x in isosceles triangle ABC The figure for Step 5 indicates the following: x = BC = 2/√3 + (1 + 1√3) = 1 + 3/√3 = 1 + √3. The correct answer is B. _________________ Mitch Hunt Private Tutor for the GMAT and GRE GMATGuruNY@gmail.com If you find one of my posts helpful, please take a moment to click on the "UPVOTE" icon. Available for tutoring in NYC and long-distance. For more information, please email me at GMATGuruNY@gmail.com. Student Review #1 Student Review #2 Student Review #3 Free GMAT Practice Test How can you improve your test score if you don't know your baseline score? Take a free online practice exam. Get started on achieving your dream score today! 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