2015 AIME II Problems/Problem 11
Problem
The circumcircle of acute has center
. The line passing through point
perpendicular to
intersects lines
and
and
and
, respectively. Also
,
,
, and
, where
and
are relatively prime positive integers. Find
.
Diagram
Solution 1
Call the and
foot of the altitudes from
to
and
, respectively. Let
and let
. Notice that
because both are right triangles, and
. Then,
. However, since
is the circumcenter of triangle
,
is a perpendicular bisector by the definition of a circumcenter. Hence,
. Since we know
and
, we have
. Thus,
.
.
Solution 2
Notice that , so
. From this we get that
. So
, plugging in the given values we get
, so
, and
.
Solution 3
Using the diagram given, . Since
bisects any chord,
and
. From there,
. Thus,
. Using
, we get
. Now let's find
. Using
, ${NP=\frac{2}{5}r^2-\frac{5}{2}$ (Error compiling LaTeX. Unknown error_msg). Therefore,
.
.
See also
2015 AIME II (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 |
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