**Problem Analysis:**
In this problem, we have a stone thrown from the top of a high tower at an angle of 37 degrees above the horizontal. We are asked to find the magnitude of the velocity when the stone's velocity becomes perpendicular to its initial velocity.

**Initial Velocity Components:**
To solve this problem, we first need to find the initial velocity components of the stone. The given initial speed is 12 m/s, and the angle of projection is 37 degrees above the horizontal. We can use trigonometry to find the horizontal and vertical components of the initial velocity.

The horizontal component (Vx) can be found using the equation:
Vx = V * cos(θ)
where V is the initial speed and θ is the angle of projection.

Substituting the given values:
Vx = 12 * cos(37°) = 9.6 m/s

The vertical component (Vy) can be found using the equation:
Vy = V * sin(θ)
where V is the initial speed and θ is the angle of projection.

Substituting the given values:
Vy = 12 * sin(37°) = 7.2 m/s

**Vertical Component Becomes Zero:**
Next, we need to find the time it takes for the vertical component of the velocity to become zero. We can use the equation of motion for vertical motion:

Vy = Vy0 + gt

where Vy is the vertical component of velocity, Vy0 is the initial vertical component of velocity, g is the acceleration due to gravity, and t is time.

Since the stone is thrown upwards, the initial vertical component of velocity is positive. Therefore, we can rewrite the equation as:

0 = 7.2 + (-9.8)t

Simplifying the equation, we get:

9.8t = 7.2

t = 7.2 / 9.8 = 0.735 seconds

**Horizontal Component Remains Constant:**
Since there is no horizontal force acting on the stone, the horizontal component of velocity remains constant throughout the motion. Therefore, the horizontal component of velocity after 0.735 seconds will also be 9.6 m/s.

**Final Velocity:**
To find the magnitude of the final velocity, we can use the Pythagorean theorem. The final velocity vector is the vector sum of the horizontal and vertical components:

V = √(Vx^2 + Vy^2)

Substituting the values:
V = √(9.6^2 + 0^2) = 9.6 m/s

So, the magnitude of the final velocity when the stone's velocity becomes perpendicular to its initial velocity is 9.6 m/s.

**Conclusion:**
In this problem, we found the magnitude of the final velocity when a stone thrown from the top of a high tower at an angle of 37 degrees above the horizontal has a velocity perpendicular to its initial velocity. The magnitude of the final velocity was found to be 9.6 m/s.