23 August 2007

A puzzle

Given a scalene right triangle with sides of integer length and perimeter P, show that there is a corresponding scalene triangle with one angle which is 60 degrees and perimeter 1.5P.

For example, the 3-4-5 triangle is a right triangle with perimeter 12; the 3-7-8 triangle has a 60-degree angle (the one opposite the side of length 7) and has perimeter 18.

I'll post the solution tomorrow.

(The puzzle isn't mine; it's from the puzzle column in the March/April 2007 issue of Technology Review, which I was flipping through earlier today.)

2 comments:

michael cassidy said...

I never heard the word 'scalene' before; as a kid I loved 3-45 triangles.

Blinding Insight said...

Every pythagorean triple can be generated as (2rs+s^2,
2r^2+2rs,
r^2+(r+s)^2)

Then the following triple forms a scalene triangle with a 60 degree angle:

(s^2 + 2sr,
s^2 + 3rs + 3r^2
s^2 + 4rs + 3r^2)

where the middle term is the side opposite to teh 60 degree angle.

Good puzzle!