Identify properties of triangle SVT.

We are given that $|VS| = |ST|$. This means that $\triangle SVT$ is an isosceles triangle.
In an isosceles triangle, the angles opposite the equal sides are equal.
Therefore, $\angle TVS = \angle TSV$.
We are given $\angle SVT = 48^\circ$.
So, $\angle TSV = 48^\circ$.

The Alternate Segment Theorem states that the angle between a tangent and a chord drawn from the point of contact is equal to the angle in the alternate segment.
Here, TY is the tangent at point T, and VT is a chord.
The angle between the tangent TY and the chord VT is $\angle VTY$.
The angle in the alternate segment subtended by the chord VT is $\angle VST$ (which is the same as $\angle TSV$).
From Step 1, we found $\angle TSV = 48^\circ$.
Therefore,

The value of $\angle VTY$ is $\boxed{\text{48}^\circ}$.