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505-555 (Easy)|   Geometry|      
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If CD is the diameter of the circle, does x equal 30? 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

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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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D
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If CD is the diameter of the circle, does x equal 30? 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=180-90-60=30. Sufficient.

Answer: D.
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damamikus
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If CD is the diameter of the circle, does x equal 30? 27 Feb 2014, 05:52
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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.

Answer: D.
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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
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If CD is the diameter of the circle, does x equal 30? 27 Feb 2014, 06:21
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damamikus
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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.

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 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
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In a triangle the ratios of the sides and the ratios of the angles not necessarily equal to each other.
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If CD is the diameter of the circle, does x equal 30? 03 Apr 2021, 08:06
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Hi - If we have two side ratios of a Pythagorean triple and one angle (e.g., the 90 degree angle, given the triangle is inscribed in the circle), can we always definitively say that it's a Pythagorean triple every time?
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If CD is the diameter of the circle, does x equal 30? 03 Apr 2021, 09:20
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samgyupsal
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Hi - If we have two side ratios of a Pythagorean triple and one angle (e.g., the 90 degree angle, given the triangle is inscribed in the circle), can we always definitively say that it's a Pythagorean triple every time?
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If you know that two sides of a right triangle are say 3 and 4, then the third side can be 5 (so in this case you'd have a Pythagorean triplet) OR \(\sqrt{7}\) (so in this case you wouldn't have a Pythagorean triplet). Therefore knowing the lengths of only two sides of a right triangle is not enough to find the third one, unless of course you know the lengths of two specific sides, meaning that you know say the length of a hypotenuse and the length of one of the legs OR the length of two non-hypotenuse legs.

P.S. Whether a triangle is inscribed in a circle is irrelevant in this case.
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If CD is the diameter of the circle, does x equal 30? 13 Apr 2021, 11:46
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want to know how did conclude that if CD = 2x and BD= x. the third side HAS to be √3 x; i.e a 30-60-90 triangle. the third side can be anything between 2x-x< third side< 2x+x. this is the only reason I was dubious about B being the write answer, as I could see everything pointing towards a 30-60.90, or is it the case, in a right-angled triangle, where hypt is twice of the other side, does the triangle must be a 30-60-90?

similarly, can we say for a rt. angle triangle with side ratio: √2*(side) : 1*side : <Unknown side> MUST be a 45-45-90 triangle.
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If CD is the diameter of the circle, does x equal 30? 13 Apr 2021, 11:54
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ueh55406
Bunuel

want to know how did conclude that if CD = 2x and BD= x. the third side HAS to be √3 x; i.e a 30-60-90 triangle. the third side can be anything between 2x-x< third side< 2x+x. this is the only reason I was dubious about B being the write answer, as I could see everything pointing towards a 30-60.90, or is it the case, in a right-angled triangle, where hypt is twice of the other side, does the triangle must be a 30-60-90?

similarly, can we say for a rt. angle triangle with side ratio: √2*(side) : 1*side : <Unknown side> MUST be a 45-45-90 triangle.
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If a hypotenuse is twice one of the legs, then we have 30°-60°-90° triangle, where the sides are in the ratio \(1 : \sqrt{3}: 2\).
Hypotenuse is 2x and one of the legs is x, then \(x^2 + y^2 = (2x)^2\) --> \(y = x\sqrt{3}\)

If a hypotenuse is \(\sqrt{2}\) times one of the legs, then we have 45°-45°-90° triangle, where the sides are in the ratio \(1 : 1: \sqrt{2}\).
Hypotenuse is \(x\sqrt{2}\) and one of the legs is x, then \(x^2 + y^2 = (x\sqrt{2})^2\) --> \(y = x\)
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If CD is the diameter of the circle, does x equal 30? 22 Oct 2023, 15:23
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