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What is the value of x?

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by AAPL » Sat Jan 20, 2018 10:24 am
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If a circle is circumscribed around a quadrilateral, as shown above, what is the value of x?

A. 105
B. 110
C. 115
D. 120
E. 125

The OA is C.

I don't have clear this PS question, I appreciate if any expert explain it for me. Thank you so much.
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by EconomistGMATTutor » Sat Jan 20, 2018 12:40 pm
If a circle is circumscribed around a quadrilateral, as shown above, what is the value of x?

A. 105
B. 110
C. 115
D. 120
E. 125

The OA is C.

I don't have clear this PS question, I appreciate if any expert explain it for me. Thank you so much.
Hi AAPL,
Let's take a look at your question.

We know that the in a circle when two chords of a circle make an angle at the circumference of the circle, the arc subtended by that angle is always twice the angle.
Therefore, in the circle given,
$$2x\ =\ arclength\ \left(BAD\right) ... (i)$$
$$2\times65\ =\ arclength\ \left(BCD\right)$$
$$\ 130=\ arclength\ \left(BCD\right ... (ii))$$

Adding Eq(i) and (ii),
$$\ 2x+130=\ arclength\ \left(BAD\right)+\ arclength\ \left(BCD\right)$$
$$\ 2x+130=\ 360$$
$$\ 2x=\ 360-130$$
$$\ 2x=\ 230$$
$$\ x=\ \frac{230}{2}=115$$

Therefore, Option C is correct.

Hope it helps.
I am available if you'd like any follow up.
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by [email protected] » Sat Jan 20, 2018 12:41 pm
Hi AAPL,

This question is built around one specific Geometry rule: when a quadrilateral is inscribed in a circle, opposite angles sum to 180 degrees. Thus, angle X =180 - 65 = 115 degrees.

Final Answer: C

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