If the area of the circle above with center O is 64π, what is the area

Question

https://gmatclub.com/forum/download/file.php?id=47686 If the area of the circle above with center O is 64π, what is the area of triangle LMN? (1) x = 2y (2) OL = LM 2019-04-30_1241.png

sumi747 · Answer

From statement 1 we can get the value of angles of the triangle and hence determine the area
From statement 2 we can get the lengths of the sides of the triangle using Pythagoras theorem and hence determine the area
Therefore
IMO D

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SaquibHGMATWhiz · Answer

Solution:      
  Pre Analysis:  
  Given area of the circle  =64\pi  
 Or  \pi r^2=64\pi 
 ⇒r^2=64 
 ⇒OL=ON=r=8  
 We are asked the area of triangle LMN

Statement 1:   x = 2y
  We already know  x+y=90  and  x=2y  means  x=60    and  y=30  
 So,  triangle LMN is a 30-60-90 triangle and we know side  LN=8+8=16   
 After which we can find the length of other sides ( 1:\sqrt{3}:2 ) and get the area  =\frac{1}{2}	imes MN	imes ML  
 Thus,  statement 1 alone is sufficient and we can eliminate options B, C and E

Statement 2:   OL = LM
  Accordig to this statement,  LM=OL=8  
 We already know  LN=8+8=16  
 We can find the length of MN using Pythagoras theorem and get the area  =\frac{1}{2}	imes MN	imes ML  
 Thus,  statement 2 alone is also sufficient

Hence the right answer is    Option D

ScottTargetTestPrep · Answer

Question Stem Analysis:

We need to determine the area of triangle LMN, which is inscribed in circle with center O.

We are told that the area of the circle is 64 \pi , which means we can solve  \pi r^2 = 64 \pi  to find that the radius of the circle is 8. Since LN is a diameter of the circle, the length of LN is 2(8) = 16.

Since the interior angles of a triangle must add up to 180, we can write x + y + 90 = 180, which is equivalent to x + y = 90.

Statement One Alone:

\Rightarrow  x = 2y

Substituting x = 2y in x + y = 90, we obtain 2y + y = 3y = 90, which means y = 30. Thus, the triangle LMN is a 30-60-90 right triangle. Since we know the length of the hypotenuse of triangle LMN, we can determine the lengths of LM and MN using 30-60-90 right triangle ratios. Since we can determine the lengths of LM and MN, we can determine the area of the triangle LMN. Statement one alone is sufficient.

Eliminate answer choices B, C, and E.

Statement Two Alone:

\Rightarrow  OL = LM

Draw the radius OM. Since OL and OM are both radii of the circle, they have the same length. Since OL = LM, all sides of the triangle OLM have the same length, which means that OLM is an equilateral triangle. Thus, x = 60 degrees. It follows that triangle LMN is a 30-60-90 right triangle, and we can proceed as in the analysis of the previous statement to determine the area of triangle LMN. Statement two alone is sufficient.

Answer: D

If the area of the circle above with center O is 64π, what is the area

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GMAT Data Sufficiency (DS) Questions

If the area of the circle above with center O is 64π, what is the area

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