2021 AIME I Problems/Problem 11
Problem
Let be a cyclic quadrilateral with
and
. Let
and
be the feet of the perpendiculars from
and
, respectively, to line
and let
and
be the feet of the perpendiculars from
and
respectively, to line
. The perimeter of
is
, where
and
are relatively prime positive integers. Find
.
Solution
Let be the intersection of
and
. Let
.
Firstly, since , we deduce that
is cyclic. This implies that
, with a ratio of
. This means that
. Similarly,
. Hence
It therefore only remains to find
.
From Ptolemy's theorem, we have that . From Brahmagupta's Formula,
. But the area is also
, so
. Then the desired fraction is
for an answer of
.
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 10 |
Followed by Problem 12 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.