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

Home > Shapes
Two Digit Diameter Trial (Posted on 2016-07-11) Difficulty: 3 of 5
Consider a circle, where the length of diameter PQ is a 2-digit integer. Reversing the digits of PQ one obtains the length of a perpendicular chord RS.

PQ intersects RS at the point T. The center of the circle is denoted by O and it is known that the length of TO is a positive rational number.

Determine the length PQ.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
Solution analytic solution Comment 2 of 2 |
At the heart of this problem is the Pythagorean theorem:
or since OR=PQ/2


Let PQ=10a+b so RS=10b+a
So 11 must be a factor of (a²-b²) and any other factor must be a perfect square.
Checking the three possibilities:
(a²-b²)=99 not possible since a<10
(a²-b²)=44 not possible (easy to check)
(a²-b²)=11 yields a=6, b=5
so PQ=65

  Posted by Jer on 2016-07-11 17:24:07
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (0)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (3)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information