Difference between revisions of "Mock Geometry AIME 2011 Problems/Problem 1"
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Now we have <math> BE=BA\cdot\tan\angle EAB=1\cdot\tan30^\circ=\frac{\sqrt{3}}{3} </math>. Finally, <math> [ABE]=\frac{1}{2}\cdot1\cdot\frac{\sqrt{3}}{3}=\frac{\sqrt{3}}{6} </math>, and our answer is <math> 2-2\left(\frac{\sqrt{3}}{6}\right)=\frac{6-\sqrt{3}}{3} </math>, and <math> a+b+c=\boxed{012} </math>. | Now we have <math> BE=BA\cdot\tan\angle EAB=1\cdot\tan30^\circ=\frac{\sqrt{3}}{3} </math>. Finally, <math> [ABE]=\frac{1}{2}\cdot1\cdot\frac{\sqrt{3}}{3}=\frac{\sqrt{3}}{6} </math>, and our answer is <math> 2-2\left(\frac{\sqrt{3}}{6}\right)=\frac{6-\sqrt{3}}{3} </math>, and <math> a+b+c=\boxed{012} </math>. | ||
+ | |||
+ | ==Video Solution by Punxsutawney Phil== | ||
+ | https://youtube.com/OEkt8HbUVbQ |
Revision as of 22:04, 23 December 2022
Problem
Let be a unit square, and let
be its image after a
degree rotation about point
The area of the region consisting of all points inside at least one of
and
can be expressed in the form
where
are positive integers, and
shares no perfect square common factor with
. Find
Solution
Denote the intersection of and
as
. Also, denote a polygon enclosed in square brackets as the area as that polygon. (For example,
denotes the area of polygon
.)
The area of the union of the two squares is equal to . Note that
, so we wish to find
.
Draw in . Notice that
, since
,
, and
, so the triangles are congruent by HL. Thus, their areas are equal, and we need to find
.
Note that , since that's the angle of rotation, so
. Also, since
, we have
.
Now we have . Finally,
, and our answer is
, and
.