# Can a triangle be formed with these length constraints?

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1. Oct 28, 2016

### Nipuna Weerasekara

1. The problem statement, all variables and given/known data
There is a triangle with sides $$3,3r,3r^2$$ such that 'r' is a real number strictly greater than the Golden Ratio.
Is this statement true or false...?
2. Relevant equations
$$Golden \space Ratio = \phi = 1.618...$$
3. The attempt at a solution
Actually I have no clue at all how to approach to this kind of question.

2. Oct 28, 2016

### Math_QED

When I give you 3 line segments with a random length, can you make a triangle with it?

3. Oct 28, 2016

### Nipuna Weerasekara

I guess so

4. Oct 28, 2016

### micromass

Staff Emeritus
Take a paper, try to make a triangle with sides 1cm, 1cm and 10cm. Show us the triangle you found.

5. Oct 28, 2016

### Nipuna Weerasekara

Actually there is a theorem called triangle inequality theorem, which states given ABC triangle, which has a,b,c length sides a+b>c , a+c>b and b+c>a must be true. therefore no need to show it to you that I cant create a triangle out of those sides 1,1 and 10.

6. Oct 28, 2016

### Nipuna Weerasekara

Actually this gives me an idea how to approach the question.
i can try to get a result for r by solving three inequalities
$$3r^2 +3 > 3r$$ $$3r^2+3r>3$$ $$3+3r>3r^2$$

7. Oct 28, 2016

### Nipuna Weerasekara

Eventually one inequality leads that r has a complex variance. (by solving 3r^2 +3 > 3r)

8. Oct 28, 2016

### micromass

Staff Emeritus
Is,'t that trvially satisfied?

9. Oct 28, 2016

### Nipuna Weerasekara

I don't get what you say here.

10. Oct 28, 2016

### Nipuna Weerasekara

But in a sense I can say that r is not strictly a real number hence the statement is false... Am I right???

11. Oct 28, 2016

### Nipuna Weerasekara

No I was wrong all along... I solved it. The statement is true. All r is real and greater than Golden Ratio (1.61803)...

12. Oct 28, 2016

### Staff: Mentor

So $r=10$ is obviously greater than $\phi$ and you've found a triangle with sides $3, 30$ and $300$?
Which curvature do you assume your geometry has?