If x, y and z represent the number of faces, number of vertices and number of edges respectively of a polyhedr, then which of the following is true?

Options

A. x+2=z-y
B. z-2=x-y
C. y-2=z+x
D. x-2=z-y

Answer

To solve the question regarding the relationship between the number of faces (x), vertices (y), and edges (z) of a polyhedron, we will use Euler's formula for polyhedra, which states:

\[ V - E + F = 2 \]

Where:
- \( V \) is the number of vertices (y)
- \( E \) is the number of edges (z)
- \( F \) is the number of faces (x)

### Step-by-Step Solution:

1. **Identify the Variables**:
- Let \( x \) represent the number of faces.
- Let \( y \) represent the number of vertices.
- Let \( z \) represent the number of edges.

2. **Write Euler's Formula**:
According to Euler's formula, we have:
\[ y - z + x = 2 \]

3. **Rearrange the Formula**:
We can rearrange this equation to express it in terms of one variable:
\[ x + y - z = 2 \]

4. **Interpret the Result**:
This equation shows the relationship between the number of faces, vertices, and edges of a polyhedron.

5. **Conclusion**:
Therefore, the relationship that holds true for a polyhedron is:
\[ x + y - z = 2 \]