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Let's go over each question step by step to provide the correct solutions.

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### **1. Problem 1: Distance from A to line CE**
- **Given:** Cube with edge length 4 cm.  
- **Question:** Find the shortest distance from **A** to the line **CE**.

**Solution:**  
The shortest distance between a point and a skew line can be found using the geometry of the cube. Here, the shortest path is along the face diagonal between points on the perpendicular faces containing the point A and the line CE.  
- Using symmetry, the required distance can be calculated as:
  
  \[ \text{Distance} = \frac{1}{2} \times \text{Face Diagonal} = \frac{1}{2} \times 4\sqrt{2} = 2\sqrt{2} \, \text{cm} \]

Answer: **C. \(2\sqrt{2}\)** cm

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### **2. Problem 2: Distance from F to line AC (edge length = 10 cm)**
- **Given:** Cube with edge length 10 cm.  
- **Question:** Find the shortest distance from **F** to the diagonal **AC**.

**Solution:**  
The point F lies on the face **EFGH**. The shortest distance from a point to a diagonal on the opposite face can be calculated as:

  \[ \text{Distance} = \frac{a\sqrt{3}}{2} = \frac{10\sqrt{3}}{2} = 5\sqrt{3} \, \text{cm} \]

Answer: **E. \(5\sqrt{3}\)** cm

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### **3. Problem 3: Distance from point K (intersection of EG and FH) to line BG**
- **Given:** Edge length = 6 cm.  
- **Question:** Find the shortest distance from **K** (intersection point of diagonals **EG** and **FH**) to **BG**.

**Solution:**  
Since K is the midpoint of space diagonal intersections, we compute the shortest distance by symmetry. It will be half the length of the space diagonal:

  \[ \text{Space diagonal} = a\sqrt{3} = 6\sqrt{3} \]
  \[ \text{Distance} = \frac{6\sqrt{3}}{2} = 3\sqrt{3} \, \text{cm} \]

Answer: **C. \(3\sqrt{6}\)** cm

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### **4. Problem 4: Distance from G to line BD (edge length = 8 cm)**
- **Solution:**  
The shortest distance is also calculated using spatial relations:

  \[ \text{Distance} = \frac{8\sqrt{2}}

The user submitted an image of a series of math problems related to calculating distances within a cube.

Explore solutions to geometry problems involving distance calculations within a cube. Learn how to find distances between points and lines, using properties of 3D coordinates and cube geometry, suitable for high school students.

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