Refer to the figure above, is triangle ABC an equilatera

Question

https://gmatclub.com/forum/download/file.php?id=54208 Refer to the figure above, is triangle ABC an equilateral triangle? (1) ∠ BCD = 60 DEG (2) D is the midpoint of AC Capture.JPG

GMATBusters · Answer

(1) ∠ BCD = 60 DEG
only one angle is known,  Not sufficient.

(2) D is the midpoint of AC
 
   Concept    The altitude to the base of an isosceles triangle bisects the base.  
 If the altitude to the base of a triangle bisects the base, it is an isosceles triangle.  
 https://gmatclub.com/forum/download/file.php?id=54223 
(PS: the Naming of vertices is different from the question) 
hence the triangle ABC is an isosceles triangle. But  INSUFFICIENT  To prove that it is an Equilateral triangle.

Combining St 1 & 2 , 
ABC is an isosceles triangle. with sides AB= BC
hence the angle ∠ BCD = ∠ BAD = 60 deg.
so, all angles = 60
 it is an Equilateral triangle.

Sufficient

Answer C

GMATWhizTeam · Answer

Solution 
 Step 1: Analyse Question Stem 
 • Let us redraw the given diagram,
  https://www.dropbox.com/s/grxokn5jvuvgcon/equilateral_DS.png?dl=1 
• ∠ ADB =  ∠CDB  = 90  degrees
 o So,  w + x = 90  Degrees
o And  y + z = 90  Degrees  
Now, we need to find if triangle ABC is equilateral.
  o Thus, we need to figure out 
  if  w = z = (x+y)= 60  degrees
 Or,  AB = BC = AC    
 Step 2: Analyse Statements Independently (And eliminate options) – AD/BCE 
 Statement 1:  ∠  BCD = 60  DEG
 • According to this statement,  w = 60  degrees
 o So,  x = 90 – 60 = 30  degrees. 
• However, we still don’t know the value of z or y. 
Hence, statement 1 is NOT sufficient and we can eliminate answer Options A and D.
 Statement 2:  D is the midpoint of AC
 • According to this statement,  AD = CD ,
• Since in right angled triangles ABD and CBD, 
 o  AD = CD 
o BD is common.
o So, AB = BC
  Hence,  w = z   
• Thus, triangle ABC is an isosceles triangle. However, we don’t whether AC is equal to AB and BC or not. So we cannot conclusively say if ABC is an equilateral triangle. 
Hence, statement 2 is also NOT sufficient and we can eliminate answer Option B.
 Step 3: Analyse Statements by combining. 
 • From statement 1:  w = 60  degrees
• From statement 2:  w = z 
• On combining the two statements, we get,
 o  w = z = 60  degrees 
• Also, we know that in triangle ABC,  w + (x+y) + z = 180 
 o So,   x+y = 60 
o Therefore,  w = (x+y) = z   
Thus, the correct answer is  Option C.

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Refer to the figure above, is triangle ABC an equilatera

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