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The problem provides a 3D geometry setup with a pyramid-like structure \( TABC \), where \( TA = AB = AC = 6 \, \text{cm} \), and it is stated that \( TA \), \( AB \), and \( AC \) are mutually perpendicular. The question asks for the distance between point \( A \) and the plane \( TBC \).

### Step-by-Step Breakdown:

1. **Shape Structure**: Since \( TA \), \( AB \), and \( AC \) are perpendicular to each other, the points \( T \), \( A \), \( B \), and \( C \) form a right-angled tetrahedron. The three edges originating from \( A \) are mutually perpendicular, making the figure geometrically simple to work with.

2. **Plane Equation for \( TBC \)**: To find the distance from point \( A \) to the plane \( TBC \), we need the equation of the plane \( TBC \). Given the symmetry, we can use the coordinates of points \( T \), \( B \), and \( C \) to find the normal vector to the plane.

   - Assume the coordinates are:
     - \( T(6, 0, 0) \)
     - \( B(0, 6, 0) \)
     - \( C(0, 0, 6) \)

3. **Distance from Point to Plane**: The general formula for the distance from a point \( P(x_1, y_1, z_1) \) to a plane \( Ax + By + Cz + D = 0 \) is:

   \[
   d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}
   \]

   The plane \( TBC \) can be defined by the normal vector derived from \( \overrightarrow{TB} \) and \( \overrightarrow{TC} \). Given the symmetry, the equation of the plane simplifies to:

   \[
   x + y + z = 6
   \]

4. **Substitute Point \( A(0, 0, 0) \)** into the equation:

   Using the point-to-plane distance formula, the distance from point \( A(0, 0, 0) \) to the plane \( x + y + z = 6 \) is:

   \[
   d = \frac{|0 + 0 + 0 - 6|}{\sqrt{1^2 + 1^2 + 1^2}} = \frac{6}{\sqrt{3}} = 2\sqrt{3}
   \]

### Final Answer:

The distance from point \( A \) to the plane \( TBC \) is \( 2\sqrt{3} \, \text{cm} \).

Would you like further details or have any questions about this solution? Here are 5 related questions:

1. How do you derive the equation of a plane given three points?
2. What is the significance of mutually perpendicular edges in geometry?
3. How do you find the normal vector to a plane in 3D space?
4. What are other applications of the point-to-plane distance formula?
5. How would the solution change if the dimensions were different?

**Tip**: Always remember that perpendicular edges in a geometric figure help simplify the calculations using coordinate geometry.

Diketahui bidang empat T. ABC dengan TA=AB=AC=6 cm, dimana TA, AB dan AC saling tegak lurus. Jarak antara titik A ke bidang TBC adalah berapa cm?

Learn how to find the distance between point A and the plane TBC in a 3D perpendicular tetrahedron with sides TA=AB=AC=6 cm. This problem involves applying the point-to-plane distance formula, suitable for students in Grades 10-12.

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