All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Shapes
Bat wings (Posted on 2005-06-08) Difficulty: 3 of 5

In the picture all of the red segments are of equal length and all of their endpoints lie on one of the two blue lines. Determine all possible values for the smaller angle between the blue lines.

(This problem is a special case of a problem discovered by Daniel Shapiro, professor of mathematics at Ohio State University. His problem generalizes a problem he saw years ago.)

See The Solution Submitted by McWorter    
Rating: 4.3333 (3 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
re(2): Label the figure (Solution) | Comment 5 of 14 |
(In reply to re: Label the figure by McWorter)

Let x=COB = FOE
Let y=ACB = DFE = CAB = FDE
Let each red segment equal 1 = AB,BC,CD,DE,EF,FA
then CBO = y-x and BD=2cos(y-x)
also ABO = 180-x-y and BF = 2cos(180-x-y) = -2cos(x+y)
FD = 2sin(90-y) = 2cosy
and since BF+FD=BD
-2cos(x+y) + 2cosy = 2cos(y-x)
-cos(x+y) + cosy = cos(y-x)
-cosxcosy + sinxsiny + cosy = cosycosx + sinysinx
-cosxcosy + cosy = cosycosx
cosy = 2cosxcosy
and as long as cosy is not 0 (y isn't 90)
1 = 2cosx
x = Arccos 1/2 = 60

Edited on June 9, 2005, 4:00 pm
  Posted by Eric on 2005-06-09 15:58:17

Please log in:
Login:
Password:
Remember me:
Sign up! | Forgot password


Search:
Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (8)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2024 by Animus Pactum Consulting. All rights reserved. Privacy Information