Mock AIME 3 Pre 2005 Problems/Problem 9
Problem
is an isosceles triangle with base
.
is a point on
and
is the point on the extension of
past
such that
is right. If
and
, then
can be expressed as
, where
and
are relatively prime positive integers. Determine
.
Solution 1
Let the foot of the altitude from to
be point
and from
to
be point
Since
is isosceles,
so
Then
Notice that
corresponds to
,
corresponds to
and
corresponds to
by some scale factor
. Also note that
Then we have
Then
is
Solution 2 (fakesolve, don't do this)
Let the perpendicular from intersect
at
Let
intersect
at
Then let
intersect
at
Note that with a factor of
So
and
Then the mass of
is
and the mass of
is
and the mass of
is
Because the triangle is isosceles, the mass of
is also
So
See Also
Mock AIME 3 Pre 2005 (Problems, Source) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |