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

Home > Shapes
Area Ascertainment (Posted on 2014-02-06) Difficulty: 3 of 5
A piece of paper has the precise shape of a triangle ABC with two side lengths AB and AC being respectively 36 and 72 with ∠ABC = 90o

Find the area of the set of points P inside ΔABC such that if creases are made by folding (and then unfolding) each of A, B, C to P, then the creases do not overlap.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution solution Comment 1 of 1

By playing with Geometers' Sketchpad we can see that point P must lie within a semicircle with diameter BC and also within a semicircle with diameter AB. It also must lie within the semicircle with diameter AC, but, this, being the hypotenuse, encompasses the whole triangle.

The two important semicircles meet at B and at a point--call it D--along the hypotenuse AC, so we need to find the combined areas of two back-to-back segments of circles with endpoints B and D.

The segment with arc centered at the midpoint of BC has area pi*(18*sqrt(3))^2 / 6 - 27*9*sqrt(3).

The segment with arc centered at the midpoint of AB has area pi*18^2 / 3 - 9*9*sqrt(3).

That brings the total area to

pi*18^2*5/6 - 324*sqrt(3)

= 270*pi - 324*sqrt(3) ~= 287.045554816926

 


  Posted by Charlie on 2014-02-06 16:36:27
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 (4)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Chatterbox:
Copyright © 2002 - 2017 by Animus Pactum Consulting. All rights reserved. Privacy Information