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В кубе ABCDA1B1C1D1ABCDA_1B_1C_1D_1, ребро которого 23\sqrt{23}, найдите расстояние от точки AA до плоскости BDA1BDA_1

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Q. В кубе ABCDA1B1C1D1ABCDA_1B_1C_1D_1, ребро которого 23\sqrt{23}, найдите расстояние от точки AA до плоскости BDA1BDA_1
  1. Cube Vertex Distance: The distance from a vertex to the opposite face in a cube is the same as the distance from any vertex to its opposite face, which is the space diagonal of the cube.
  2. Calculate Space Diagonal: To find the space diagonal, we use the formula for the diagonal of a cube, which is 3\sqrt{3} times the length of one edge.
  3. Find Edge Length: The edge length is given as 23\sqrt{23}, so we calculate the space diagonal as 3×23\sqrt{3} \times \sqrt{23}.
  4. Perform Multiplication: Perform the multiplication to find the space diagonal: 3×23=(3×23)\sqrt{3} \times \sqrt{23} = \sqrt{(3 \times 23)}.
  5. Simplify Square Root: Simplify the expression under the square root: 3×23=69\sqrt{3 \times 23} = \sqrt{69}.
  6. Final Result: So, the distance from point AA to plane BDA1BDA_1 is 69\sqrt{69}.

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