Answer
An isosceles triangle has at least two sides with the same length.
Since $AB=BC=\sqrt{13}$, then points $A$, $B$, and $C$ are vertices of an isosceles triangle.
Work Step by Step
Step 1. Given $A(3,4),B(1,1),C(-2,3)$, we need to show that at least two sides are equal in length.
Step 2. Determine the length of side $\overline{AB}$ using the distance formula:
$AB=\sqrt {(3-1)^2+(4-1)^2}=\sqrt {13}$
Step 3. Determine the length of side $\overline{BC}$ using the distance formula:
$BC=\sqrt {(-2-1)^2+(3-1)^2}=\sqrt {13}$
Since $\triangle{ABC}$ has two congruent sides then it is an isosceles triangle.
Thus, points $A$, $B$, and $C$ are vertices of an isosceles triangle.
