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Integer Side Lengths (Posted on 2009-03-05) Difficulty: 2 of 5
The side lengths of a triangle are consecutive integers.
The largest interior angle is twice the smallest.

What are the side lengths of the triangle?

  Submitted by Bractals    
Rating: 3.0000 (1 votes)
Solution: (Hide)
Let the side lengths be n-1, n, and n+1. Let the smallest angle be x.
The smallest angle is opposite n-1 and the largest opposite n+1.
From the sine rule we get
        n-1            n            n+1
      -------- = ------------- = ---------     
       sin(x)     sin(180-3x)     sin(2x)

        n-1          n          n+1
  ==> -------- = --------- = ---------   
       sin(x)     sin(3x)     sin(2x)

        n-1               n                   n+1
  ==> -------- = -------------------- = ---------------   
       sin(x)     sin(x)[4cos2(x)-1]     2sin(x)cos(x)

        n-1           n            n+1
  ==> -------- = ------------ = ---------   
         1        4cos2(x)-1     2cos(x)

                 n+1                        2n-1
  ==> cos(x) = --------   and   cos2(x) = -------- 
                2(n-1)                     4(n-1)

  ==> n(n-5) = 0
Therefore, the side lengths are 4, 5, and 6.

See "complete solution by Daniel for a solution using the cosine rule.

See "Geometric solution!" by Chesca for a solution using similar triangles.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
Puzzle AnswerK Sengupta2022-07-24 03:22:05
re: To Jer and a ConfirmationJer2009-03-06 13:24:51
re: Geometric solution!Jer2009-03-06 13:15:03
Geometric solution!Chesca Ciprian2009-03-06 06:03:19
To Jer and a Confirmationbrianjn2009-03-06 03:33:34
re: Different solutionbrianjn2009-03-06 03:24:51
Different solutionJer2009-03-05 14:02:01
Solutioncomputer solutionCharlie2009-03-05 12:48:10
Solutioncomplete solutionDaniel2009-03-05 12:22:46
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