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16 Sep 2014, 00:49
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Points \(A\), \(B\), \(C\) and \(D\) lie on a circle of radius 1. If \(x\) is the length of arc \(AB\) and \(y\) the length of arc \(CD\) respectively, such that \(x \lt \pi\) and \(y \lt \pi\), is \(x \gt y\)?
(1) \(\angle ADB\) is acute
(2) \(\angle ADB \gt \angle CAD\)
(1) \(\angle ADB\) is acute
(2) \(\angle ADB \gt \angle CAD\)
16 Sep 2014, 00:49
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Official Solution:
Statement (1) by itself is insufficient.
Statement (2) by itself is insufficient. These angles seem to define the lengths of arcs. The angles can come from different sides of the arc. On the images below you can see that both cases comply with the S2, but have different answers to the question. We need one more condition to define the arcs.
Statements (1) and (2) combined are sufficient. If \(\angle\) ADB is acute and greater than \(\angle\) CAD, then arc \(AB\) is greater than arc \(CD\).
Answer: C
Statement (1) by itself is insufficient.
Statement (2) by itself is insufficient. These angles seem to define the lengths of arcs. The angles can come from different sides of the arc. On the images below you can see that both cases comply with the S2, but have different answers to the question. We need one more condition to define the arcs.
Statements (1) and (2) combined are sufficient. If \(\angle\) ADB is acute and greater than \(\angle\) CAD, then arc \(AB\) is greater than arc \(CD\).
Answer: C
General Discussion
