If segment BD has a measure of x which of the following expresses the

Question

https://gmatclub.com/forum/download/file.php?id=51274 If segment BD has a measure of x which of the following expresses the perimeter of the triangle ABC in terms of x? A. 6x + 2x\sqrt{3} B. 6x + \frac{2x\sqrt{3}}{3} C. 4x + 4x\sqrt{3} D. 3x + x\sqrt{3} E. 2x + 2x\sqrt{3} image027.jpg

Kinshook · Answer

If segment BD has a measure of x which of the following expresses the perimeter of the triangle ABC in terms of x?
Angle BDC = 180 - 30 - 60 = 90
Angle BAD = 180 - 90 - 60 = 30
BD = x

In triangle BDC:
sin 60 = BD/BC ;  BC = x /sin 60 = 2x/\sqrt{3} 
tan 60 = BD/CD;  CD = x /tan 60 = x/\sqrt{3}

In triangle BAD:
sin 30 = BD/AB; AB = x/sin 30 = 2x
tan 30 = BD/AD;  AD = x/tan 30 = \sqrt{3}x

Perimeter of triangle ABC = AB + BC + CA = AB + BC + CD + DA =  2x + 2x/\sqrt{3} + x/\sqrt{3} + \sqrt{3}x = (2 + \sqrt{3} + 3/\sqrt{3})x = 2x + 2x\sqrt{3}

IMO E

GMATWhizTeam · Answer

Solution:  
 • In triangle BCD,
 o  ∠BDC = 90 
o BCD is a 30: 60 : 90 triangle
o  CD: BD: BC = 1: \sqrt{3}: 2 
   \frac{BD}{BC} = \frac{\sqrt{3}}{2} 
  \frac{x}{BC} = \frac{\sqrt{3}}{2} 
  BC = \frac{2x }{ \sqrt{3}} = \frac{2*\sqrt{3}*x}{3}    
 • In triangle ABC,
 o  ∠ABC = 90, ∠BCA = 60, ∠BAC = 30 
o ABC is a 30: 60: 90 triangle
o  BC: AB: AC = 1: \sqrt{3}: 2 
o  \frac{BC}{AB} = \frac{1}{\sqrt{3}} 
   \frac{2*\sqrt{3}*x}{3*AB} = \frac{1}{(\sqrt{3})} 
  AB = 2*x  
o  \frac{BC}{AC} = \frac{1}{ 2} 
   \frac{2*\sqrt{3}*x}{3*AC} = \frac{1 }{ 2} 
  AC = \frac{4*x*\sqrt{3}}{3}   
• Perimeter of triangle =  AB + BC + AC = 2*x + \frac{2*x*\sqrt{3}}{3} + \frac{4*\sqrt{3}*x}{3} = 2x + \frac{6*x*\sqrt{3}}{3} = 2x + {2*x*\sqrt{3}  
Hence, the correct answer is  Option E.

GMATGuruNY · Answer

In a 30-60-90 triangles, the sides are in the following ratio:
1:√3:2

Let CD=1, as in the figure below:

https://imagehost.imageupload.net/2020/04/08/three-30-60-90-triangles.png

In the resulting figure, x=BD=√3 and the perimeter of triangle ABC= 6+2√3.

Which of the following expresses the perimeter of the triangle ABC in terms of x? 
When x=√3, the correct answer = 6+2√3.
Only E works:
2x + 2x√3 = 2√3 + 2(√3 * √3) = 2√3 + 6

E

ScottTargetTestPrep · Answer

We see that triangle BCD is a 30-60-90 triangle. Since BD = x, DC is:

DC * √3 = x

DC = x/√3

Thus, BC = 2 * DC = 2x/√3

Now, we see that triangle ABC is also a 30-60-90 triangle. Since BC is opposite the 30-degree angle and AB is opposite the 60-degree angle, AB = BC * √3 = 2x/√3 * √3 = 2x. Similarly, since AC is opposite the 90-degree angle, AC = BC * 2 = 2x/√3 * 2 = 4x/√3.

Thus, the perimeter of triangle ABC is:

AB + BC + AC = 2x + 2x/√3 + 4x/√3 = 2x + 6x/√3 = 2x + 6x√3/3 = 2x + 2x√3

Answer: E

Vidhi96 · Answer

In an triangle with 60-30-60, the ration of length of hypotenuse :Long length : short length is 2x : xroot3 : x - Hint

If segment BD has a measure of x which of the following expresses the

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If segment BD has a measure of x which of the following expresses the

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