2021 JMPSC Invitationals Problems/Problem 12
Contents
[hide]Problem
Rectangle is drawn such that
and
.
is a square that contains vertex
in its interior. Find
.
Solution 1 (Clever Construction)
We draw a line from to point
on
such that
. We then draw a line from
to point
on
such that
. Finally, we extend
to point
on
such that
.
Next, if we mark as
, we know that
, and
. We repeat this, finding
, so by AAS congruence,
. This means
, and
, so
. We see
, while
. Thus,
~Bradygho
Solution 2 (Trig)
Let . We have
, and
. Now, Law Of cosines on
and
gets
and
, so
~ Geometry285
Solution 3 (Mass points and Ptolemy)
Let be the center of square
. Applying moment of inertia to the system of mass points
(which has center of mass
) gives
Since
is a right triangle, we may further cancel out some terms via Pythag to get
To compute
, apply Ptolemy to cyclic quadrilateral
(using the fact that
is 45-45-90) to get
. Thus
~djmathman
See also
- Other 2021 JMPSC Invitationals Problems
- 2021 JMPSC Invitationals Answer Key
- All JMPSC Problems and Solutions
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