Refer to the figure above, is triangle ABC an equilatera

Refer to the figure above, is triangle ABC an equilateral triangle?
(1) ∠ BCD = 60 DEG
(2) D is the midpoint of AC

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Attachment:

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Solution

Step 1: Analyse Question Stem

• Let us redraw the given diagram,

• ∠\(ADB = \)∠CDB\( = 90\) degrees

o So, \(w + x = 90\) Degrees
o And \(y + z = 90\) Degrees

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

• 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.

• 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.

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, \(w = (x+y) = z\)