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

Home > Just Math
Concentric Circles and Equilateral Triangle (Posted on 2018-12-15) Difficulty: 4 of 5
Three concentric circles each have radii 8, 15 and 17.

Find the largest possible area of the equilateral triangle with one vertex on each circle.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 5.0000 (1 votes)

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution To be exact Comment 2 of 2 |

Construction:

Construct Circle(8), radius 8 on the origin, with centre O.

Point A is at {0,8}.

Point P is at {-7.5,0} Construct a line through P parallel to the y axis. Construct Circle(15), radius 15 on O. By Pythagoras, the line through P intersects Circle(15) at B in the 3rd quadrant at {-7.5,-(15 sqrt(3))/2)}

Point Q is on the line through P, 4 above B. Construct a line through Q parallel to the x axis. Construct Circle(17), radius 17 on O. By Pythagoras, the line through Q intersects Circle(17) at C in the 4th quadrant at {sqrt(17^2-(((-15 sqrt(3))/2)+4)^2),(-(15 sqrt(3))/2)+4)}, simplifying nicely to {(1/2 (15 + 8 sqrt(3)),(-(15 sqrt(3))/2)+4)}, around {-14.42,-8.99}

ABC is the desired equilateral triangle: sqrt((1/2 (15 + 8 sqrt(3)))^2+((((15 sqrt(3))/2)-4)+8)^2) = sqrt(7.5^2 + (8 + 1/2 (15 sqrt(3)))^2) = sqrt(4^2+(7.5+(1/2 (15 + 8 sqrt(3))))^2

From line AB {0,8}{-7.5,-(15 sqrt(3))/2)}, the side length of the triangle is sqrt(7.5^2 + (8 + 1/2 (15 sqrt(3)))^2), so its area is:

sqrt(3)*(sqrt(7.5^2 + (8 + 1/2 (15 sqrt(3)))^2))^2/4, or more simply:

1/4 (360 + 289 sqrt(3))

Update: there is a much neater way of doing this, well worth a more general problem, which I have now added to the queue.


Edited on December 16, 2018, 3:33 am
  Posted by broll on 2018-12-15 23:11:11

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 (11)
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