Question

What is the maximum area of a rectangle that can be inscribed in a triangle of area S ?

Answer 1

Any pictures or diagrams?

Answer 2

None given

Answer 3

but this should be it

Answer 4

Hmmm...Yes, I should think it would be that in some form or another.

A problem I have is that the tri might be isosceles, equilateral, or some bizarrely skewed shape.

I'll have to stare at this for awhile and see what I can come up with. :-)

Answer 5

Interesting,

I drew an isosceles with a base of 6 and a height of 4 which gives an area of 12 sq units.

The largest rectangle I could make inside it was 6 sq. units which I made by drawing a line parallel to the base and halfway up, giving it a height of 2 and a width of 3, which gives 6 sq units.

So it looks like for an isosceles of area S, the largest rectangle I can fit inside it has an area of 1/2 S.

Answer 6

To wit... :-)

Answer 7

How can you assume it to be isosceles ?

Answer 8

I can't.

However, I was just fiddling with this one, too, and it looks like the rectangle is still half the area of the triangle.

The triangle is 9 x 3.555 which is 15.9975 sq units.
The rectangle is 1.778 x 4.5 which is 8.001 sq units.

Rounding errors may account for what looks like ought to be 16 units v. 8 units.

Answer 9

yeah...
(9 * 3.55555555555555555555555555555)/2 is effectively exactly 16
(1.777777777777777777777 * 4.5) is effectively exactly 8

Answer 10

for a triangle, its comes out to be one third but :|

Answer 11

right triangle*

Answer 12

I dunno...I'm still getting a 1:2 ratio

Here's a right triangle with area = 16, and a rectangle contained with an area = 8

Answer 13

In this drawing, you can notice that the fatter rectangle isn't as spacious as the more square-like rectangle. In fact, the greatest area you can get is from having your rectangle as a square where length=width.
You'll notice that in order for your rectangle to be a square, it divides the two segments of your triangle into halves!

Help at all?