Can this rectangle fit in an isosceles triangle with two known sides?

Question

Could a rectangle, 120cm by 60cm, fit inside a triangle with two sides that are 203cm each? (Third side unknown) Edit: (I'm not a math student. This is a problem I encountered in life and figured I'd use this opportunity to learn something new)

Answer 1

Hint: Draw a picture. There are two configurations that make sense. The triangle is isosceles. It may help to bisect the triangle to make two right triangles and try to fit half the rectangle in. Let $x$ be the leg of the right triangle. If $x$ is too short the triangle is not wide enough for the rectangle. If $x$ is too long, the triangle is not tall enough. Is there a happy medium?

Answer 2

Some more configurations worth looking at:

Put two vertices on one of the length-203 sides, with one at or near the vertex between them. This comes out particularly simply if it's a right triangle at that vertex.

OK, the obtuse angle isn't going to be optimal - reducing the angle to $90^\circ$ will make things easier in that configuration. In the acute-angle range, there's a tradeoff, so it isn't obvious what angle is best just by looking.