AIME II 2010 #12
Posted:
Wed May 05, 2010 2:57 amby stupidityismygam
Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is [unparseable or potentially dangerous latex formula]. Find the minimum possible value of their common perimeter.
Re: AIME II 2010 #12
Posted:
Fri May 07, 2010 5:48 pmby stupidityismygam
Let one isosceles triangle have sides of [unparseable or potentially dangerous latex formula] and the other have sides of [unparseable or potentially dangerous latex formula].
Then, we have that [unparseable or potentially dangerous latex formula].
Now looking at the areas we have that
[unparseable or potentially dangerous latex formula][unparseable or potentially dangerous latex formula]. Since [unparseable or potentially dangerous latex formula], we can simplify further to [unparseable or potentially dangerous latex formula].
Substituting for [unparseable or potentially dangerous latex formula] ([unparseable or potentially dangerous latex formula]), we get that [unparseable or potentially dangerous latex formula]. Thus, minimum perimeter will be obtained when [unparseable or potentially dangerous latex formula]