2015 AMC 12A Problems/Problem 20
Problem
Isosceles triangles and
are not congruent but have the same area and the same perimeter. The sides of
have lengths
,
, and
, while those of
have lengths
,
, and
. Which of the following numbers is closest to
?
Solution
The area of is
and the perimeter is 18.
The area of is
and the perimeter is
.
Thus , so
.
Thus , so
.
We square and divide 36 from both sides to obtain , so
. This factors as
. Because clearly
but
, we have
The answer is
.