If CD is the diameter of the circle, does x equal 30?

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If CD is the diameter of the circle, does x equal 30? [#permalink]

05 Feb 2012, 17:08

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73% (01:22) correct

26% (01:03) wrong

based on 3 sessions

If CD is the diameter of the circle, does x equal 30?

(1) The length of CD is twice the length of BD.
(2) y = 60

Attachment:

Triangle.jpg [ 13.63 KiB | Viewed 1162 times ]

This is how I am trying to solve this, but there is bit of a guess work. So can someone please help?

Considering the figure CD is the diameter of the circle and its the hypotenuse of the triangle too i.e. Angle CBD= 90 degrees. --------------------------------------(1). This is where I am guessing.

If that's the case then considering statement 1

Knowing that the side ratios of the 30:60:90 degree triangle are 1:\sqrt{3}:2 the know that the x = 30 and y = 60 as the x is the angle opposite to the shortest leg. Therefore, sufficient.

Statement 2

x + y + B = 180
x+60+90 = 180
x = 30. Sufficient.

[Reveal] Spoiler: OA

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Re: Is angle x = 30 degrees? [#permalink]

05 Feb 2012, 18:01

Attachment:

Triangle.jpg [ 13.63 KiB | Viewed 1156 times ]

If CD is the diameter of the circle, does x equal 30?

A right triangle inscribed in a circle must have its hypotenuse as the diameter of the circle. The reverse is also true: if the diameter of the circle is also the triangle’s hypotenuse, then that triangle is a right triangle.

So, angle CBD is a right angle.

(1) The length of CD is twice the length of BD --> ratio of a hypotenuse to a side is 2:1 --> we have 30°, 60°, and 90° right triangle, where the sides are always in the ratio 1:\sqrt{3}:2. BD corresponds with 1, thus it's smallest side and opposite the smallest angle (30°). Sufficient.

(2) y = 60. x=180-90-60=30. Sufficient.

Answer: D.
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Re: Is angle x = 30 degrees? [#permalink]

05 Feb 2012, 18:05

enigma123 wrote:

If CD is the diameter of the circle, does x equal 30?

(1) The length of CD is twice the length of BD.
(2) y = 60

This is how I am trying to solve this, but there is bit of a guess work. So can someone please help?

Considering the figure CD is the diameter of the circle and its the hypotenuse of the triangle too i.e. Angle CBD= 90 degrees. --------------------------------------(1). This is where I am guessing.

Hi,You are right in your logic. In fact what you are guessing is actually true ,with respect to the figure-- If the hypotenuse of the triangle is also the diameter of the circle , then the angle opposite to it is a right angle .
In other words 'the angle inscribed by the diameter of a circle is a right angle'.
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