Isosceles Triangle Length?

$\text{In geometry, an isosceles triangle is a triangle that has two sides of equal length.}$

given that, AB=1 & BC=3

so, $A C$ could be both $1 o r 3$

but according to the rule of triangles, $\text{"The sum of any two side lengths of a triangle will always be greater than the third side length"}$

so, $\text{AC must be 3, otherwise that will be 1+1=2<BC}$

ABC is isosceles triangle where AB=1 and BC=3.

According to the rule of Triangle, "The sum of any two side lengths of a triangle will always be greater than the third side length".

So the possible value of AC will be satisfied the following inequality,

BC-AB<AC<AB+BC

=> 3-1<AC<1+3

=> 2<AC<4

So the value of AC can be between 2 to 4(Exclusive)

As ABC is isosceles triangle, so AC should be 1 or 3.

But According to the inequality, AC not equal 1

So, AC=3

We apply the triangle inequality to answer this. First, because $ABC$ is an isosceles triangle, $AC=1$ or $AC=3$

Case 1: $AC=1$

$AB+BC> AC \Rightarrow AB+AC>BC$

But $\displaystyle 1+1 < 3$ , so $AC=1$ is not the correct answer.

Case 2: $AC=3$

$AB+BC> AC \Rightarrow AB+AC>BC$

We could see that $\displaystyle 1+3 > 3$

Therefore, $AC=\boxed{3}$