2022 AMC 10B Problems/Problem 20
Problem
Let be a rhombus with
. Let
be the midpoint of
, and let
be the point
on
such that
is perpendicular to
. What is the degree measure of
?
Solution (Law of Sines and Law of Cosines)
Without loss of generality, we assume the length of each side of is 2.
Because
is the midpoint of
,
.
Because is a rhombus,
.
In , following from the law of sines,
We have .
Hence,
By solving this equation, we get .
Because ,
In , following from the law of sines,
Because , the equation above can be converted as
Therefore,
Therefore, .
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
Solution 2
Extend segments and
until they meet at point
.
Because , we have
and
, so
by AA.
Because is a rhombus,
, so
, meaning that
is a midpoint of segment
.
Now, , so
is right and median
.
So now, because is a rhombus,
. This means that there exists a circle from
with radius
that passes through
,
, and
.
AG is a diameter of this circle because . This means that
, so
, which means that
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)