2019 AMC 10B Problems/Problem 16
Problem
In with a right angle at
, point
lies in the interior of
and point
lies in the interior of
so that
and the ratio
. What is the ratio
Solution
Without loss of generality, let and
. Let
and
. As
and
are isosceles,
and
. Then
, so
is a 3-4-5 triangle with
.
Then , and
is a 1-2-
triangle.
On isosceles triangles and
, drop altitudes from
and
onto
; denote the feet of these altitudes by
and
respectively. Then
by AAA similarity, so we get that
, and
. Similarly we get
, and
.
-scrabbler94