In hexagon ABCDEF, AB=3, BC=4, CD=5, DE=6 and AF=7

Hello IanStewart

That was a brilliant explanation
Thank you so much

Can you help to resolve my query about the minimum length in this case?

You don't exactly need to calculate as such. If you calculate sum of rest 5 sides 3+4+5+6+7 = 25
in order for one of the side of hexagon be 25.
other lines has to form a straight line and will coincide with the line segment 25.
so 25 or more than 25 can't be the side of hexagon.
values lesser than 25 are hypothetically possible.
Hence Answer is C

Agreed with the maximum value part
But what if the options had a value less than 5, which as per egmat explanation would be wrong

Hence I want to understand how the minimum value needs to be calculated

see the highlighted part
in triangle AEF,
one side is 7 other is less than 18
so the maximum length of third side is indeed less than 7+18 = 25
but the minimum length is more than 18-7 = 11
so the constraint is
11<EF<25 not 2<EF<25.

I'm not going to read their explanation, because it's inordinately complicated, but if they're concluding there's some minimum value here, the explanation is mathematically incorrect. Any value between 0 and 25 is possible.

There sometimes could be a relevant minimum value. I'll use a quadrilateral instead of a hexagon for simplicity. If you had a quadrilateral with sides 1, 2, 5 and x, then since the sum of any three sides must exceed the fourth, we know x must be less than 1+2+5, so must be less than 8. But in this case (and this happens because the numbers are a bit far apart in this example), there is also a minimum value of x, because if we look at the three smallest sides, 1, 2 and x, their sum must exceed the fourth side of 5. So x must be greater than 2 in this case. But in your original hexagon question, there is no minimum value for the sixth side, because no matter what length it has (less than 25), the sum of any five sides will automatically be greater than the sixth side.

This concept is tested most often with triangles, incidentally, and with triangles specifically, you'll almost always need to consider both minimum and maximum values. So if you have sides of 3, 4, and y in a triangle, then y must be less than 7 (since 3+4 > y must be true), but must be greater than 1 (since 3 + y > 4 must also be true). Any value in that range is possible though.

I did actually read the explanation, and it doesn't make any sense. Just looking at the last line: suppose you know that 2 < AE < 18. You also know that AF is 7. We want to know the minimum possible length for the third side EF of this triangle. The minimum is not 5, which I gather they are finding by minimizing AE (they seem to be assuming AE is slightly larger than 2). That's just logically wrong, because if you make AE longer, you can make EF smaller. If AE is 7, for example, so our triangle has sides of length 7, 7 and x, the minimum value of x is anything greater than zero. It clearly is not 5.

IanStewart

Thank you so much for your valuable insights on this question!

I see IanStewart has already resolved your query brilliantly (the way he always does!) so I will not add anything regarding the solution to this question.
I will give you a diagram though which most of my students find very useful in understanding this concept.

Say you have a side of length 5. To make a triangle, you need two more sides. If the other sides add up to 5, they will just overlap on this side as shown in the first diagram. To make a triangle the two sides will need to add up to a bit more than 5 and you can keep going up.



Now everything stems from this concept. If you know that one side of a hexagon is 10, put it down as the base and try to make a hexagon on top of it. Obviously the sum of all other sides will be more than 10 so they don't all overlap on the base. Does it matter how many sides the polygon has? No. Always, the sum of all other sides will be greater than any one side.

Now what happens when one side is unknown - the 'x'

I put x down and say that its length must be less than the sum of all other sides - makes sense. So I get the maximum value of x.
But also, same is true for all other sides of the polygon too. So if I put down the next largest side (7 in the original question), the sum of the rest of the sides (including x) must be greater than 7 too. Since sum of 3+4+5+6 is already greater than 7, I don't have to worry about the value of x and it can take any value.

Say if the given sides were 1, 2, 3, 4 and 12. Now x must be less than 1+2+3+4+12 = 22.
But also sum of all other sides must be greater than 12 (the next greatest side). Note that 1+2+3+4 add up to 10 only so x must be more than 2. So x here has a minimum value too.

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