arjtryarjtry wrote:
Points A, B, C and D lie on a circle of radius 1. Let x be 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 > \angle CAD
The answer is not B. If we use only that statement, we can draw the following (no need to use co-ordinate geometry, but it's easier to explain if I do, since I can't draw a picture):
-draw the circle in the co-ordinate plane, with centre at (0,0). The radius is 1.
-make AC a diameter: put A at (-1, 0), and C at (1,0).
-draw D and B in the third quadrant (i.e. where x and y are both negative), and so that when read counterclockwise, the points are in the sequence ADBC. That is, minor arc AD should be shorter than minor arc AB.
Drawn this way, ADB should be (very) obtuse, and much larger than CAD, which must be less than 90 degrees (angle CDA must be 90, since AC is a diameter so CAD and DCA are both less than 90). The length of minor arc AB is clearly less than one quarter of the circumference, while minor arc CD is clearly greater than one quarter of the circumference, so it is possible for x to be less than y.
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