The Natural Triangle

Geometry Level 4

The sides of a triangle are three natural consecutive numbers and its largest angle is twice the smallest one. Find the sum of sides.


The answer is 15.

Sanjeet Raria by Sanjeet Raria , 27, India

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2 solutions

Michael Mendrin
Jun 19, 2014

The proof of this should be self-evident with this diagram

456 Triangle 456 Triangle

2 Helpful 0 Interesting 0 Brilliant 0 Confused

But how did these numbers come into your mind?

Sanjeet Raria - 6 years, 11 months ago

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By brute force, of course, using the Cosine Law. I was a little surprised that an integer solution even exists, so I was sure there was a simpler explanation, and thus this diagram. This is a perfect example of what my father had framed up in his office wall, "This problem, once solved, becomes simple."

Using this diagram, a less brutish force method of finding the sides should be possible, if the sides are unknown.

Michael Mendrin - 6 years, 11 months ago

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Your skills are just awesome!!

Sanjeet Raria - 6 years, 11 months ago

Actually my solution was a long one. Still wondering about the beauty of simplicity.

Sanjeet Raria - 6 years, 11 months ago

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@Sanjeet Raria – Later this afternoon let's see if I can do it again in the other way.

Michael Mendrin - 6 years, 11 months ago

I used the Sine Law instead. With sides x, x+1 and x+2 and respective opposite angles A, (180 - 3A) and 2A we have the equations sin(A) / x = sin(2A)/(x + 2) and sin(A) / x = sin(3A)/(x + 1). Using the double- and triple-angle sine formulae I quickly found that x = 2/(2*cos(A) - 1), and then upon substitution the value cos(A) = 3/4 dropped out quite nicely, giving me x = 4, making the other side lengths 5 and 6, leading to a final answer of 15.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

I did pretty same.

Sanjeet Raria - 6 years, 11 months ago

Assume triangle ABC with AB=(x-1),BC=x & AC=(x+1). Draw BP the angle bisector of B meeting AC in P. Now / C=/ PBC=/_ABP. With the angle bisector theorem and the fact that triangles ABP and ABC are similar, one can set up the equation: (x^2-1)(x+1)=(x-1)^2*(2x-1) which yields x=0, 1 or 5 of which only x=5 is admissible. Hence the perimeter=15. Neither any brute force nor any trigonometry is required.

Ajit Athle - 6 years, 11 months ago

I first used sine rule which gave me x+2/x = cos smallest theta then, I used the cosine rule for the smallest side and thus gave me the equation of cos smallest theta = x+5/2(x+2)

And as they are both equal to the same thing(cos smallest theta), they must be equal as well.

Therefore, x+5/2(x+2) = x+2/2x

(x+2)^2 = x^2 + 5x => x^2 + 4 + 4x = x^2 +5x => x = 4 making the other sides 5, 6 and hence the sum is equal to 15

This is quite simple and I am considering to post it as a solution. How did you do @Krishna Ar?

Kartik Sharma - 6 years, 11 months ago

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@Kartik Sharma - Where did you learn sine rule and cosine rule from? And how are you so awesome at calculus? CAN you tell me a good place to do advanced limits?

Krishna Ar - 6 years, 11 months ago

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@Krishna Ar(these tags don't work for me :) I had some books and from there I learnt it. And yes, the 2nd question hahaha AWESOME at calculus???? who told you this? By the way, you are a level-4 at Calculus and me, I am just a level-3 -_- By the way, you are really too awesome being level 4-5 in almost all the subjects(except 2) You are a secret!!! By the way, did you give JOMO and if yes how much are you scoring?

Kartik Sharma - 6 years, 11 months ago

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@Kartik Sharma – hEY..../u ARE FAR BETTER THAN ME IN CALC ...i CAN ASSURE THAT! WHICH BOOK WAS THT? (SINE RULE)...u see...u have a lot of good books and thus u study from it and are awesome!. What about JOMO...did u ugive it....i didnt @Kartik Sharma

Krishna Ar - 6 years, 11 months ago

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