Maximum length of a side in a quadrilateral

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Max Bacon

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Jul 21, 2014
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Here's the problem:

The lengths of the sides of a quadrilateral are all integers. If three of the sides have lengths of 3, 4 and 5, what is the maximum length of the fourth side?

I know it's not 12, but then again I do not know what it is. Any help is appreciated.
 
Max Bacon said:
Here's the problem:

The lengths of the sides of a quadrilateral are all integers. If three of the sides have lengths of 3, 4 and 5, what is the maximum length of the fourth side?

I know it's not 12, but then again I do not know what it is. Any help is appreciated.
Click to expand...

The limiting distance would be if all the vertices of the quadrilateral were in a straight line, but then it wouldn't be a quadrilateral would it? So the question becomes, IMO, (1) does the fourth side actually have an attainable maximum and, if not, (2) does the length have a least upper bound and, if so, what is it.
 
Max Bacon said:
The lengths of the sides of a quadrilateral are all integers. If three of the sides have lengths of 3, 4 and 5, what is the maximum length of the fourth side?

I know it's not 12, but then again I do not know what it is. Any help is appreciated.
Click to expand...
How do you "know it's not 12"? What was your reasoning?

Use that same reasoning to determine the maximal integer length. ;)
 
Ishuda said:
The limiting distance would be if all the vertices of the quadrilateral were in a straight line, but then it wouldn't be a quadrilateral would it? So the question becomes, IMO, (1) does the fourth side actually have an attainable maximum and, if not, (2) does the length have a least upper bound and, if so, what is it.
Click to expand...

stapel said:
How do you "know it's not 12"? What was your reasoning?

Use that same reasoning to determine the maximal integer length. ;)
Click to expand...

I realised it's not 12 because it would be too obvious. Then when Ishuda mentioned a straight line I thought that 3+4+5 would give a straight line, not a side. So the biggest value, that will still keep the figure a quadrilateral, should be 11?
 
Yes, if the three segments, of length 3, 4, and 5, were put end to end in a single line then you would have length 12. That is not a quadrilateral but if you tilt the two end segments a very, very small angle to the middle segment, you can have a quadrilateral with fourth side of length as close to 12 as you please.

Since you are required to have integer lengths, you can keep "tilting" the two end segments to get a fourth side length of "11".
 
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