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I think that circle geometry questions can be some of the toughest, because a lot of times you have to be pretty creative. They also often involve triangles, even when they do not explicitly set it up for you like in this question.

This problem focuses heavily on the inscribed angle rule of circles. The inscribed angle rule is cousin to the central angle rule, and you treat inscribed angles the same as central angles but you double them. This means that the given 30 degree inscribed angle will etch out the equivalent of a 60 degree central angle between B and C. This means that the arcs from D to B on the left and from C to D on the right will each be the other 150 degrees of the circle (360 - 60 = 300, 300/2 = 150). Using the inscribed angle rule again, we see that x is an inscribed angle to the entire arc from B to C and C to D, or 60+150 = 210. Given that X is an inscribed angle and not a central angle, we use the rule in reverse and divide 210 by 2 in order to get X = 105 degrees, or answer choice E. Being able to reverse engineer rules (use rules backwards) is key to solving tricky gmat questions.

The inscribed angle rule is less well known than the central angle rule, so the gmat will use it to separate test takers. Make sure you are solid on all fundamental rules of geometry! Veritas does an awesome job of highlighting these rules (which are necessary but definitely not sufficient to crushing the gmat), and then delving deeply into strategic use of the rules.

Re: If BD=DC, what is the value of x in the figure above? [#permalink]

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24 May 2015, 10:21

I assumed that because both lines BD and CD are equal length that they were both diameters of the circle. Then, according to Thale's Theorem (http://en.wikipedia.org/wiki/Thales%27_theorem) I could work with 90 degree angles. Is this correct or did I just luckily come across the correct answer? Thanks in advance.

I assumed that because both lines BD and CD are equal length that they were both diameters of the circle. Then, according to Thale's Theorem (http://en.wikipedia.org/wiki/Thales%27_theorem) I could work with 90 degree angles. Is this correct or did I just luckily come across the correct answer? Thanks in advance.

That is not correct, so I am guessing that you somehow got lucky on this one. We actually know that BD and CD are definitely both not diameters of the circle, because a diameter of the circle has to run through a circle's center, and these two lines are not intersecting anywhere near the center. Because they are the same length and are both chords of the circle, and a diameter is the longest chord of a circle, we know that neither of them are diameters.

I am curious as to how you would get the correct answer assuming that they are both diameters? If BD is a diameter, then Thale's Theorem would make x 90 degrees, not 105 degrees which is the correct answer.

Re: If BD=DC, what is the value of x in the figure above? [#permalink]

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21 Sep 2016, 11:12

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Re: If BD=DC, what is the value of x in the figure above? [#permalink]

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21 Sep 2016, 12:14

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Note: Angle subtended on the circumference is always taken from the ARC and NOT the segment. I had done ~ to Ian but instead of using 210 angle subtended by arc BCD i used angle 150 thinking that as angle x is drawn from segment BD we have to make 150 half. That is not the case. We always take into consideration the arc on the OPPOSITE side of angle (central or inscribed in both cases). Hopefully the image helps:

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Re: If BD=DC, what is the value of x in the figure above? [#permalink]

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19 Nov 2017, 05:23

BuggerinOn wrote:

I assumed that because both lines BD and CD are equal length that they were both diameters of the circle. Then, according to Thale's Theorem (http://en.wikipedia.org/wiki/Thales%27_theorem) I could work with 90 degree angles. Is this correct or did I just luckily come across the correct answer? Thanks in advance.

VeritasPrepBrandon It's the property of cyclic quadrilaterals that opposite angles add up to 180 degrees. Since we have 75 opposite to x, x = 105. (D)
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