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If CD is the diameter of the circle, does x equal 30? [#permalink]
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05 Feb 2012, 17:08
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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.
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.
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Re: Is angle x = 30 degrees? [#permalink]
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05 Feb 2012, 18:01
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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=1809060=30. Sufficient. Answer: D.
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Re: Is angle x = 30 degrees? [#permalink]
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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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Re: If CD is the diameter of the circle, does x equal 30? [#permalink]
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05 Feb 2012, 18:10
Agree dentobizz and Bunuel  but no where in question says that its a diameter  so are we assuming or am I missing something?
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Re: If CD is the diameter of the circle, does x equal 30? [#permalink]
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05 Feb 2012, 18:16
enigma123 wrote: If CD is the diameter of the circle, does x equal 30?
CD is the diameter of the circle as given in the question stem
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Re: If CD is the diameter of the circle, does x equal 30? [#permalink]
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05 Feb 2012, 18:16
My sincere apologies. I agree and thanks to both of you.
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Re: Triangle Inscribed [#permalink]
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10 Jan 2013, 12:29
D
If CD is a diameter then triangle BCD is a right angle triangle with angle B = 90.
For triangle BCD, angle B +x +y = 180
Therefore we know that x+y=90.
1) . CD = 2 x BD.
By Pythagorus Theorem, CD x CD = (BD x BD) + (BC x BC)
=> (BCxBC) = 4 (BDxBD)  (BDxBD) = 3 (BDxBD) => BC = \sqrt{3} BD
Tan x = BD / BC = BD/(\sqrt{3}BD) = 1/\sqrt{3}
=> x = 30
SUFFICIENT
2) y=60
& x+y=90
=> x= 9060 = 30
SUFFICIENT



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Re: Is angle x = 30 degrees? [#permalink]
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27 Feb 2014, 05:52
Bunuel wrote: Attachment: Triangle.jpg 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=1809060=30. Sufficient. Answer: D. I am having difficulties applying ratios in triangles. If CD=2BD then the their ratio is (CD/BD)= 2. Based on this, shouldn't the ratio of their corresponding angles (90° corresponds to side CD, and x° corresponds to BD) be the same? So, (90°/x°)=2 > x°=45° I know this is wrong, and I understand the explanation using the 306090 ratio but I don't understand why my ratio lead to the wrong solution. I hope someone can clarify this for me cheers, Max



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Re: Is angle x = 30 degrees? [#permalink]
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27 Feb 2014, 06:21
damamikus wrote: Bunuel wrote: Attachment: Triangle.jpg 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=1809060=30. Sufficient. Answer: D. I am having difficulties applying ratios in triangles. If CD=2BD then the their ratio is (CD/BD)= 2. Based on this, shouldn't the ratio of their corresponding angles (90° corresponds to side CD, and x° corresponds to BD) be the same? So, (90°/x°)=2 > x°=45° I know this is wrong, and I understand the explanation using the 306090 ratio but I don't understand why my ratio lead to the wrong solution. I hope someone can clarify this for me cheers, Max In a triangle the ratios of the sides and the ratios of the angles not necessarily equal to each other.
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Re: If CD is the diameter of the circle, does x equal 30? [#permalink]
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16 Sep 2016, 06:00
Another, simple explanation why (1) is sufficient:
CB is an inscribed chord with a length equalling the radius of the circle. Every point on the circle has an equal distance (i.e. the radius) to the center. Drawing a triangle OCB (O being the center) reveals that OC=CB=OB and <OCB = 60 degrees. 1806090=30 for the angles of the big triangle. (1) is sufficient.



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Re: If CD is the diameter of the circle, does x equal 30? [#permalink]
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20 Sep 2016, 13:12
Definitely vote D.
Initial statement lets us know we are dealing with a right triangle
(1) SUFFICIENT  angle that is opposite 2x with be 2x the angle remaining  Thus we know we are dealing with a 306090 triangle
(2) SUFFICIENT  we know two angles and thus we can solve for the third



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