“What is the least possible perimeter of a triangle whose sides have different positive integers to its largest side ?”
The meaning is slightly ambiguous, but I’ll have a go. I assume you are not talking about a right angled triangle, as you do not mention that. If that is wrong please repost.
Rest your elbows wide apart on a table. The distance between your elbows can represent the largest side of a triangle, of length x say, as a base. The triangle is completed by where your arms cross. The lower your arms are angled, the less the perimeter, until your arms lay flat in the extreme case, and this limiting value of the perimeter is just 2x.
An example would be a base of 5 with different positive integer sides 2 and 3; but, of course as the sides would have to lay flat the shape would no longer be a triangle.
To cope with the “ different positive integers to its largest side” condition we have to add 1
The answer for the general value of the perimeter P is 2x + 1
An example would be a base of 5 with different positive integer sides 2 and 4 which has a perimeter of (2 x 5 ) + 1.
But perhaps the emphasis of the question is aimed at
“What is the least possible perimeter of any triangle of all triangles whose sides have different positive integers to its largest side ?”
We can start by considering that 1 is the smallest integer and work up from an equilateral triangle with unit sides, (reading the triangle as base, side, side).
1, 1, 1 does not fit the condition that base is longer.
2, 1, 1 is flat and not a triangle.
2, 1, 2 does not fit the condition that base is longer.
3,1, 1 does not form a triangle
3, 1, 2 is flat and not a triangle.
3, 2, 2 is OK (assuming the sides are allowed to be the same as each other while being different to the base).
If this is what is meant, the answer is perimeter = 7
If the sides are not allowed to be the same I think the minimum would be with 4, 2, 3
If this is what is meant, the answer is perimeter = 9
(N.B. These both fit our earlier general condition of P = 2x + 1
But with x = 3 if sides allowed to be same and x = 4 if not)
I hope this is the sort of answer you needed,
Regards - Ian
