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

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Attachment: Triangle.jpg [ 13.63 KiB | Viewed 14172 times ]
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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Attachment: Triangle.jpg [ 13.63 KiB | Viewed 14128 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.

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

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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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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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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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My sincere apologies. I agree and thanks to both of you.
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Re: Triangle Inscribed  [#permalink]

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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= 90-60 = 30

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

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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=180-90-60=30. Sufficient.

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 30-60-90 ratio but I don't understand why my ratio lead to the wrong solution. I hope someone can clarify this for me cheers,

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

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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=180-90-60=30. Sufficient.

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 30-60-90 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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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. 180-60-90=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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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 30-60-90 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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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=180-90-60=30. Sufficient.

Hey Bunuel, the question doesn't say that the triangle BCD is inscribed in the circle - so Point B could be anywhere.

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

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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, Tan 1/root 3 = 30 degree
=> x = 30

SUFFICIENT

2) y=60

& x+y=90

=> x= 90-60 = 30

SUFFICIENT

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

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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=180-90-60=30. Sufficient.

Bunuel
I understand that if trainagle has ratio 1:root3:2 it is 30 60 90
here we know hyp ther side is 2:1 and one angle is 90
so is this sufficent to conclude that if ratio of two sides is 2:1 and one angle is 90 it is 30 60 90 traingle ? Re: If CD is the diameter of the circle, does x equal 30?   [#permalink] 02 Aug 2019, 07:07
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