Friday puzzle -- rectangle and pentagon

Re: Friday puzzle -- rectangle and pentagon

Bill Earl

Sweet!

24 

Lee Schierer McKean, PA

Is this a trick question since you gave us the length of the longest side?

Lee

Sorry,

Alex Y

No, the 24 is the length of the longest side of the rectangular piece of paper before folding. What we are looking for is the longest side of the pentagon produced by the folded sheet.

8?

Unless

Dan Donaldson

Unless someone gets the answer, please do not post it for a few days. I want to work on it, but do not have the time right now. Good puzzle.

No 

I'm not exactly sure how to respond

Alex Y

to that request, for a reason that will become apparent later. But for now, I won't post the solution.

Glad you are enjoying it.

Re: Friday puzzle -- rectangle and pentagon

Larry Barrett

I have an answer that is also consistent with Bill's cleverly crafted answer, if I assume that Bill's answer corresponds to a current event. My answer involves a rather cumbersome proof that uses the formula(s) for the area of a rhombus.

Alex Y

Sounds like you have it as well!

Bill wins the prize for best-hidden answer.

I'll post a solution without any rhombus area formulas, after a couple of days.

Re: 

Larry Barrett

Bill also posted his answer within 30 minutes or so after you posted the puzzle, so there must be a more elegant path to the solution than looking at rhombus formulas.

Re: 

Bill Earl

My solution was based on properties of triangles.

Solution (Don't read this, Dan)

Alex Y

Bill gets a double prize, for speed of answering, and for hiding the answer so cleverly that others did not see it!

As Bill said, the key was properties of triangles. Shown below are the rectangle and pentagon. The yellow triangle is where the pentagon is two sheets thick. Since the pentagon has an area 2/3 of the rectangle, this yellow triangle must have an area of 1/3 of the total.

You should be able to convince yourself by symmetry that the two small triangles are identical (with reflection) and that the yellow triangle is isosceles (In fact, it is equilateral, but we don't know that yet.) The blue dotted line divides the yellow triangle into two right triangles, each of which have an area of 1/6 (of the rectangle) and the two small outer triangles also have an area of 1/6, Since these right triangles have the same area and share a hypotenuse, they have legs of the same length. Call the width of the rectangle "w" and the other leg of the triangles x. Then we have the area of six of these little triangles equal to the area of the rectangle. I.e., 24w = 6* (xw/2), which gives us x = 8.

So we know that the bottom of the yellow triangle is 16. Two of the other sides of the pentagon are of length 8. The other two are the width of the rectangle, and since this fold line is on an angle other than 90 degrees to the side, it is longer than w. So 16 is the length of the longest side of the pentagon.

QED

D'Oh... didn't double it!

steven antonucci

I solved for the short leg of the triangle, but should have doubled it for the longest side!

S