Gre question : finding length of sides of a triangle given the longest side length and each side has an integer length

Question

Need help in understanding the solution given for the problem in textbook. Below is the question

Below is the solution

Option F=10 is not an answer as its given that all 3 angles are different and if shortest side is 10 then the triangle will become isosceles triangle. But in above answer its states all 3 sides will become same and become equilateral. But how did it derived the 3rd side ? Similarly i could not understand other correct answers explanation eg How did it derived 7 in 4-7-10 (Choice A).

Answer 1

Suppose the shortest side is $10$ and the longest side is $10$. The remaining length must be at least as long as the shortest side and cannot exceed the longest side as well. Hence the remaining length is $10$.

As for $A,B,C,D$, we just have to check that the sum of the two shortest side exceeds the the longest side. Hence for $A$, the question is can we find an integer $4<y< 10$ such that $4+y > 10$, hence we can choose $y=7$. Similar for the other options.

Answer 2

Let the side lengths be $a < b <c$. If $a=10$ and we know $c=10$, then for $b$ to be an integer, the only possible value could be $b=10$ which would contradict that all sides have unequal lengths.

For other answers: keep in mind the triangle inequality: sum of two sides has to be bigger than the third side. So if $a=4$ and $c=10$, then with $b$ an integer to satisfy $a+b>c$, we need $4+b>10$, same as $b>6$.