cyclic quadrilateral

[gmatmachoman]

ABCD is a cyclic quadrilateral with three angles in the ratio 1 : 2 : 3. If both its diagonals are shorter than the diameter of its circumcircle, what is the measure of the smallest angle of the quadrilateral?



1. 36
2. 45
3. 60
4. 90
5. cannot be determined


[spoiler]OA :a[/spoiler]

[shovan85]

ABCD is a cyclic quadrilateral with three angles in the ratio 1 : 2 : 3. If both its diagonals are shorter than the diameter of its circumcircle, what is the measure of the smallest angle of the quadrilateral?



1. 36
2. 45
3. 60
4. 90
5. cannot be determined


[spoiler]OA :a[/spoiler]

Let the angles be x, 2x, 3x, and 360 - 6x

As the diagonal < diameter thus none of the angles are 90. (Discard option 4)
And between opposite angles one will be obtuse and other will be acute.

As ABCD is a cyclical quadrilateral then SUM(opposite angles) = 180

It is quite evident that x and 2x cannot be opposite as x comes to 60 and one angle becomes 180 and one becomes 0. Thus ABCD is no more a quadrilateral (Discard option 3)

Then consider x and 3x as opposite then x becomes 45 and 2x becomes 90. But none can be 90 as stated before. (Discard option 2)

Now, x and 360 - 6x be opposite.

x+360-6x = 180

=> x = 36. (Answer)

Alternatively,

Starting from options we can put different values of x but the only problem is we cannot say only by using options that 360 - 6x is smallest or not. Plus there is a option Cannot be determined. Definitely, not GMAT

[gmatmachoman]

Nice work Man!!!

It's definetly NOT a GMAT question. Agreed.!!