Arc and Coordinate

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Hi Experts,


Pls help me with this

[GMATGuruNY]

Given that DE is the diameter and that AB and DE are perpendicular, some test-takers will intuitively perceive the relationships shown in the figure above.
That said, here's a proof:

The diagonals of ADBE form 4 right angles.
A quadrilateral whose diagonals form 4 right angles is a kite, a rhombus, or a square.

An inscribed angle that intercepts the diameter is a right angle.
Thus, ∠DAE = 90 and ∠DBE = 90.
A rhombus with two opposite right angles is a square.
Implication:
ADBE is either a kite or a square.

In a square, the diagonals form 4 congruent triangles.
In a kite, the diagonals form 2 congruent smaller triangles and 2 congruent larger triangles.
Implication:
Whether ADBE is a square or a kite, ∆AEF and ∆BEF must be CONGRUENT.


Statement 1: arc AB = 120º
The degree measurement of an inscribed angle is 1/2 the degree measurement of the arc intercepted by the inscribed angle.
Since inscribed ∠AEB intercepts arc AB, ∠AEB = 60º.
The result is the following figure:

The figure shows that ∆ABE is EQUILATERAL, but there is no way to determine the perimeter of ∆ABE.
INSUFFICIENT.


Statement 2: AB = 2
The result is the following figure:

No way to determine the perimeter of ∆ABE.
INSUFFICIENT.


Statements combined:

Thus, the perimeter of ∆ABE = 2+2+2 = 6.
SUFFICIENT.

The correct answer is C.

[GMATinsight]

Question: Perimeter of Triangle ABC = AB+BC+AC = ?

Point to be Noted:

Since the figure is Symmetric about Diameter DE (as angle BFE = 90)
Therefore AE = BE
and AF = BF

Therefore Question rephrased

Perimeter of Triangle ABE = AB+BE+AE = (AF+FB)+AE+AE = 2AF+2AE = 2(AF+AE) = ?

Statement 1)
Measure of Arc AB = 120 [angles subtended bt arc Ab at Centre of Circle]
Property: Angle Subtended at the centre by any arc is twice the angle subtended by same arc at Circumference
Therefore angle AEB = 60
But angle AEF = Angle FEB = 60/2 = 30 (Due to symmetry about Diameter)
Therefore, Triangle AFE become a 30-60-90 Triangle with ratio of sides as 1:Sqrt3:2
But none of the sides is known therefore Insufficient

Statement 2)
AB = 2
i.e. AF = 2/2 = 1
But no property about triangle AFE is known to calculate the other sides therefore Insufficient

Combining the two Statements

Triangle AFE become a 30-60-90 Triangle with ratio of sides as 1:Sqrt3:2
AF = 2/2 = 1

Therefore, AF = 1, EF = Sqrt3 and AE = 2

Therefore perimeter of Triangle ABE = 2(AF+AE) = 2 (1+2) = 6 SUFFICIENT

Answer: Option C