A regular dodecagon (a polygon with 12 sides) is inscribes in a circle. How many diagonals of this dodecagon would not pass through the centre of the circumscribing circle?

Options

A. 24
B. 36
C. 42
D. none of these

Answer

To find the number of diagonals in a regular dodecagon (12-sided polygon) that do not pass through the center of the circumscribing circle, we can follow these steps:

### Step 1: Calculate the total number of diagonals in the dodecagon.
The formula for the number of diagonals \(D\) in a polygon with \(n\) sides is given by:

\[ D = \frac{n(n-3)}{2} \]

For a dodecagon, \(n = 12\):

\[ D = \frac{12(12-3)}{2} = \frac{12 \times 9}{2} = \frac{108}{2} = 54 \]

### Step 2: Determine how many diagonals pass through the center of the circle.
Diagonals that pass through the center of the circle connect pairs of opposite vertices. In a dodecagon, the number of pairs of opposite vertices can be calculated as follows:

Since there are 12 vertices, we can pair them up as follows:
- Vertex 1 pairs with Vertex 7
- Vertex 2 pairs with Vertex 8
- Vertex 3 pairs with Vertex 9
- Vertex 4 pairs with Vertex 10
- Vertex 5 pairs with Vertex 11
- Vertex 6 pairs with Vertex 12

This gives us a total of 6 pairs of opposite vertices, which means there are 6 diagonals that pass through the center.

### Step 3: Calculate the number of diagonals that do not pass through the center.
To find the number of diagonals that do not pass through the center, we subtract the number of diagonals that pass through the center from the total number of diagonals:

\[ \text{Diagonals not passing through center} = \text{Total diagonals} - \text{Diagonals passing through center} \]

Substituting the values we found:

\[ \text{Diagonals not passing through center} = 54 - 6 = 48 \]

### Final Answer:
The number of diagonals of the dodecagon that do not pass through the center of the circumscribing circle is **48**.

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