2018 AIME I Problems/Problem 4
Problem 4
In and
. Point
lies strictly between
and
on
and point
lies strictly between
and
on
) so that
. Then
can be expressed in the form
, where
and
are relatively prime positive integers. Find
.
Solution 1
We draw the altitude from to
to get point
. We notice that the triangle's height from
to
is 8 because it is a
Right Triangle. To find the length of
, we let
be the height and set up an equation by finding two ways to express the area. The equation is
, which leaves us with
. We then solve for the length
, which is done through pythagorean theorm and get
=
. We can now see that
is a
Right Triangle. Using this information, we set
to
and
as
.