How many diagonals can be drawn from a vertex of an octagon?
The correct answer and explanation is:
To determine how many diagonals can be drawn from a vertex of an octagon, we need to understand the basic properties of the shape and how diagonals are defined.
An octagon has eight sides and eight vertices. A diagonal in any polygon is a line segment that connects two non-adjacent vertices. From any given vertex of the octagon, a diagonal can be drawn to other vertices, except for the vertex itself and its two adjacent vertices.
So, from one vertex of the octagon, the possible vertices you can connect to (to form a diagonal) are the remaining six vertices. These six vertices are not adjacent to the given vertex, meaning that the two adjacent vertices cannot be used to form a diagonal.
Thus, from one vertex of an octagon, 6 diagonals can be drawn.
Formula for Diagonals from a Single Vertex:
To generalize, the number of diagonals that can be drawn from a vertex in any polygon with nn sides (where n≥3n \geq 3) is given by: Diagonals from a vertex=n−3\text{Diagonals from a vertex} = n – 3
This formula works because you subtract 1 for the vertex itself and 2 for the two adjacent vertices.
For an octagon, where n=8n = 8, the number of diagonals from one vertex is: 8−3=58 – 3 = 5
This means that from each vertex of an octagon, you can draw 5 diagonals.
Conclusion:
Thus, for each of the eight vertices of an octagon, you can draw 5 diagonals. This makes sense as there are a total of 8 vertices, and each vertex connects to 5 other vertices to form diagonals.