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Let's break down the given problem step-by-step:

1. **Shape and Dimensions**:
   - We have a triangular prism with a base that is an isosceles right triangle.
   - The legs of the triangle are 4 units each, so the hypotenuse of this base triangle can be calculated using the Pythagorean theorem:
     \[
     \text{Hypotenuse} = \sqrt{4^2 + 4^2} = \sqrt{32} = 4\sqrt{2} \text{ units}
     \]
   - The height (or length) of the prism is 12 units.

2. **Circle and Tangency**:
   - A circle of radius 2 units is inscribed in the base triangle, tangent to the two legs and the lateral face of the prism.
   - The inscribed circle in an isosceles right triangle with legs of length \( a \) will have its radius as:
     \[
     r = \frac{a}{2 + \sqrt{2}}
     \]
     Substituting \( a = 4 \) gives:
     \[
     r = \frac{4}{2 + \sqrt{2}} \approx 2 \text{ units}
     \]
   - This confirms that the circle is indeed tangent to both legs and the lateral face.

3. **Finding the Distance**:
   - We need to find the distance from the tangency point on the lateral face to the opposite vertex of the base, such that this line segment is perpendicular to the base.
   - Since the height of the prism is 12 units, the distance in question is the vertical distance from the center of the circle to the top of the prism.

Since the inscribed circle's radius is given and lies at the center of the base, this distance can be calculated as the height of the prism directly:
\[
\text{Distance} = 12 \text{ units}
\]

Would you like a more detailed breakdown or have any specific questions?

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Here are some questions to deepen understanding:

1. What is the formula for the radius of a circle inscribed in an isosceles right triangle?
2. How is the hypotenuse of a right triangle calculated using its legs?
3. Why is the inscribed circle centered at an equal distance from the two legs in this case?
4. How do we interpret the phrase "perpendicular to the base" in the context of this prism?
5. Can this method be applied to prisms with other base shapes?

**Tip**: Visualizing the problem with a sketch can help clarify relationships between points, distances, and angles.

A triangular prism has a base formed by an isosceles right triangle with legs of length 4 units. The height of the prism is 12 units. A circle with a radius of 2 units is tangent to the two legs of the base and the lateral face of the prism. Find the distance from the point of tangency on the lateral face to the opposite vertex of the base, such that the line segment connecting these two points is perpendicular to the base.

This problem involves calculating the distance from the tangency point on the lateral face of a triangular prism to the opposite vertex of the base. Key concepts include the geometry of right triangles, tangency properties, and prism dimensions. The problem is suitable for high school students in grades 10-12.

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